# The Mathematics of Data IAS Park City Mathematics IAS PARK CITY Mathematics 25 1470435756

# Data Mathematics (IAS/Park City Mathematics, 25) 1470435756, 9781470435752 Data Mathematics (IAS/Park City Mathematics) CITY MATHEMATICS, 25) 1470435756, 9781470435752 Data Mathematics (IAS/ Park City Mathematics) (IAS/Park City Mathematics, 25) 1470435756, 9781470435752

Table of Contents Covered Seat Women This is the introduction of lectures on random numeric lines algebra, and introduces the basics of linear algebra. standard. Vector norm. Guidance line Norm. Flobenius norm. Disassembly SVD according to peculiar value and shutten standard pseud o-inverted matrix in the basic matrix matrix. The nature of the event. Limit relevance. Random value random values for random values random values, which are independent of random values, and cumulative distribution functions with independent events and independent event conditions, are on the left. Random passenger analysis of algorithm R Scriptsize and Scriptsize Atrixm Ultique. Analyzing algorithms close to the most appropriate probability. Standard restrictions. left. Randnla approach that solves regression problems, random conversion of Hadamar. Main algorithm and main reason. Preliminary RANDNLA algorithm. Proof of theorem 5. 2. 2. Start of algorithm R Scriptsize andl Scriptsize Oosts Scriptsize QUARES. left. It is the introduction of the Randnla algorithm that is approximately a lo w-ranked matrix, B < SPAN> Contents cover title sheet, random numeric lecture algebra, and introduces the basics of linear algebra. standard. Vector norm. Guidance line Norm. Flobenius norm. Disassembly SVD according to peculiar value and shutten standard pseud o-inverted matrix in the basic matrix matrix. The nature of the event. Limit relevance. Random value random values for random values random values, which are independent of random values, and cumulative distribution functions with independent events and independent event conditions, are on the left. Random passenger analysis of algorithm R Scriptsize and Scriptsize Atrixm Ultique. Analyzing algorithms close to the most appropriate probability. Standard restrictions. left. Randnla approach that solves regression problems, random conversion of Hadamar. Main algorithm and main reason. Preliminary RANDNLA algorithm. Proof of theorem 5. 2. 2. Start of algorithm R Scriptsize andl Scriptsize Oosts Scriptsize QUARES. left. This is the introduction of the lecture on the Randnla algorithm that is approximately a lo w-ranked matrix, the B-Table cover title sheet, random numeric line algebra. standard. Vector norm. Guidance line Norm. Flobenius norm. Disassembly SVD according to peculiar value and shutten standard pseud o-inverted matrix in the basic matrix matrix. The nature of the event. Limit relevance. Random value random values for random values random values, which are independent of random values, and cumulative distribution functions with independent events and independent event conditions, are on the left. Random passenger analysis of algorithm R Scriptsize and Scriptsize Atrixm Ultique. Analyzing algorithms close to the most appropriate probability. Standard restrictions. left. Randnla approach that solves regression problems, random conversion of Hadamar. Main algorithm and main reason. Preliminary RANDNLA algorithm. Proof of theorem 5. 2. 2. Start of algorithm R Scriptsize andl Scriptsize Oosts Scriptsize QUARES. left. Randnla algorithm, B, which is approximately a lo w-ranked matrix

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\ Set the minimum square of a no n-controlled queue that optimizes the format of data analysis tasks, decomposition of spar current untouched amounts, sparing components plus, low rank rating, lo w-ranking low rank, reserve vector sub. Selection, logistic regression, no n-negative and secret research on inference for the trigger tailor conditions. Low rank of matrix, supplementary vector side selection, logistic regression, no n-controlled functions and optimal reasoning of the optimality of the optimality of the reason for the optimal function of the optimal system, a convex case, a strong convex case: linear search Methods of relative gradient Method of accelerating gradient Method of sloppy gradient Method of acceleration gradient of acceleration gradient: gentle income acceleration gradient NESTEROVA: Basic Newton La w-Newton method for convex functions, no n-fuel functions Newton Law Tertiary to Relocatio n-Restrictions by probable optimizatio n-Restriction s-Submitted area, restrictions, restrictions, restrictions, restrictions, restrictions, restrictions, restrictions < Span> Optimization of the formulation of analysis tasks, facto r-decomposition of spar current c o-amount, sparing component plus, low rank evaluation, low ranked queue sub selection, logistic regression, no n-negative and worthwhile tailor conditions Detailed research. Low rank of matrix, supplementary vector side selection, logistic regression, no n-controlled functions and optimal reasoning of the optimality of the optimality of the reason for the optimal function of the optimal system, a convex case, a strong convex case: linear search Methods of relative gradient Method of accelerating gradient Method of sloppy gradient Method of acceleration gradient of acceleration gradient: gentle income acceleration gradient NESTEROVA: Basic Newton La w-Newton method for convex functions, no n-fuel functions Newton Law Tertiary to Relocatio n-Restriction s-Restrictions by probable optimizatio n-Submitted area, restrictions, restrictions, restrictions, restrictions, restrictions, restrictions, restrictions \ Limit \ Data analysis tasks that set the minimum square of the queue of no n-contributed queues Optimization of formula, facto r-current conjugation factor, sparing component plus, lo w-ranking rating, low rank matrix low rank selection, su b-selection of spare vectors, returning to logistic, no n-negative and worthy tailor conditions the study. Low rank of matrix, supplementary vector selection, logistic regression, no n-controlled inference of smooth functions and optimality of the optimality of the optimal tailor conditions Detailed research Ather case, convex case, strong convex case: linear search Methods of relative gradient Method of accelerating gradient Method of sloppy gradient Method of acceleration gradient of acceleration gradient: gentle income acceleration gradient NESTEROVA: Basic Newton La w-Newton method for convex functions, no n-fuel functions Newton Law Tertiary Adjustment to Relocatio n-Restrictions by probability optimizatio n-Submitted area, restrictions, restrictions, restrictions, restrictions, restrictions, restrictions, restrictions

Literature such as the foundations of convex analysis, introduction and determination of the properties of convex sets, continuity and local derivatives of convex functions, and conditions of optimality of computational rules of sub-methods, The dimension of the optimality steps and metrics are introduced by Le Kam's multidimensionality and the method of Assius. Technical Applications Continuity of convex functions Probabilistic sub-method results for requirements Sabotage problems Sabotage problems and exercises introduce randomized methods of matrix computation and the main ideas of randomized approaches to rank the advantages of randomized methods Conventions from other chapters and extensive literature Decomposition according to singular values (SVD) Orthonorms Mura pseudoinversion The Penruth two-step approach is a randomized algorithm for computing matrices. Randomized algorithms of stage A "- the task of finding the bounds of the range of a single-pass algorithm Hermitian matrices General matrices - the method of complexity O (Mn Log K) for general dense matrices, the theoretical bounds of the bounds error of the Gross Probability

и двустороннего ID Детерминированные способы вычисления ID Рандомизированные способы вычисления ID Рандомизированые методы вычисления CUR- разложения CUR-разложение Переустройство двустороннего ID в CUR-разложение Адаптивное определение ранга с обновлением матрицы Постановка задачки Ненасытный метод обновления Метод блокированного обновления Оценка общепризнанных мерок остатка Адаптивное определение ранга без обновления Matrics Gerandomiseerde Methoden Voor Het Berekenen Van de Ranking QR-Kle M. Niet-up-up-inlilig subgasus fending subexponentiële schreeuwende venejkheid subgauss random vectors Lemma Johnson-lindenstraus note lecture 2. Concentrate van de h. OEVEELHEDEN NIT-WILLEKEURIGE MATRIXMATRIX ONGELIJKHEID VANSTEIN VAN Gemeenschappen 3: Evaluatie Van Covaluatie van Covariatie Evaluat IE VAN COVARIATIE EN Evaluatie Van Covariate en Evaluatie Van Covariate van Covariate van Covariate van Covariate van Covariate en evaluatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatiev OVARIATIE ENALUATIE VAN COVARIATIE ENALUATIE VAN COVARIATIE VAN COVARIATIE ENALUEREN DE MATRIXBEOREORDOORDELOODANTE MATRIXMATRIXUTRICATRITRITRITRITRITRITEITEITEIN VAN Lege Van De Matrixlezing Lecture Lecture Het Onderscheid Van de Matrix Gaussiaanse Breedte-Ingelijkheid van Verschil de MatrixoutPut Van Lemma Johnson-LindensStraus Beordel ING Van Curovy U

アリフェティック ・ シリーズ シリーズ iAS/Park City that 25

データ 数学 マイケル ・ v ・ マホーニ ・ k ・ ドゥチ ・ k ・ ギルバート レドール レドール

Literature such as the foundations of convex analysis, introduction and determination of the properties of convex sets, continuity and local derivatives of convex functions, and conditions of optimality of computational rules of sub-methods, The dimension of the optimality steps and metrics are introduced by Le Kam's multidimensionality and the method of Assius. Technical Applications Continuity of convex functions Probabilistic sub-method results for requirements Sabotage problems Sabotage problems and exercises introduce randomized methods of matrix computation and the main ideas of randomized approaches to rank the advantages of randomized methods Conventions from other chapters and extensive literature Decomposition according to singular values (SVD) Orthonorms Mura pseudoinversion The Penruth two-step approach is a randomized algorithm for computing matrices. Randomized algorithms of stage A "- the task of finding the bounds of the range of a single-pass algorithm Hermitian matrices General matrices - the method of complexity O (Mn Log K) for general dense matrices, the theoretical bounds of the bounds error of the Gross Probability

и двустороннего ID Детерминированные способы вычисления ID Рандомизированные способы вычисления ID Рандомизированые методы вычисления CUR- разложения CUR-разложение Переустройство двустороннего ID в CUR-разложение Адаптивное определение ранга с обновлением матрицы Постановка задачки Ненасытный метод обновления Метод блокированного обновления Оценка общепризнанных мерок остатка Адаптивное определение ранга без обновления Matrics Gerandomiseerde Methoden Voor Het Berekenen Van de Ranking QR-Kle M. Niet-up-up-inlilig subgasus fending subexponentiële schreeuwende venejkheid subgauss random vectors Lemma Johnson-lindenstraus note lecture 2. Concentrate van de h. OEVEELHEDEN NIT-WILLEKEURIGE MATRIXMATRIX ONGELIJKHEID VANSTEIN VAN Gemeenschappen 3: Evaluatie Van Covaluatie van Covariatie Evaluat IE VAN COVARIATIE EN Evaluatie Van Covariate en Evaluatie Van Covariate van Covariate van Covariate van Covariate van Covariate en evaluatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatie vanatiev OVARIATIE ENALUATIE VAN COVARIATIE ENALUATIE VAN COVARIATIE VAN COVARIATIE ENALUEREN DE MATRIXBEOREORDOORDELOODANTE MATRIXMATRIXUTRICATRITRITRITRITRITRITEITEITEIN VAN Lege Van De Matrixlezing Lecture Lecture Het Onderscheid Van de Matrix Gaussiaanse Breedte-Ingelijkheid van Verschil de MatrixoutPut Van Lemma Johnson-LindensStraus Beordel ING Van Curovy U

25 数学 シリーズ

データ 数学 マイケル ・ v ・ マホーニ ・ k ・ ドゥチ ・ k ・ ギルバート レドール レドール

アメリカ 数 理会話 大学 将来 な 産業 ・ 応用 数学 の 談義 談義

Life Mazseo, Michael V. Mahoni at the beginning, John K. Dachi, Anna K. Gilbert, the opening editor IAS/ Park City Mathematics Laboratory is a high school mathematician, mathematics and mathematical education researcher, mathematics teacher. , Graduate students and senior students are conducting mathematical education programs to participate in different but different research and education programs. This book includes lecture notebooks for mathematical subjects in the 2010 Summer School Program. Basic 15-02, 52-02, 60-02, 62-02, 65-02, 68-02, 90-02.

US Congress Library Catalog Inn publication Data Name Mahoney, Michael W., Editor. | Duchi, John, Editor. | | Publisher name: Mathematics Institute Park City. | Park City. Title: Data Mathematics / Michael V. MAHONI, John K. Duchi, Anna K. Gilbert, Editors. Park City Mathematics Series; Volume 25 | "Institute for Advanced Study "|" Industrial Mathematics Association "| | Includes a book magazine link. ID: lccn 2018024239 | ISBN 9781470435752 (Alk. Paper) Topics: LCSH: Mathematics-Teacher preparation-Conference. | Mathematics research and education conference | Big data conference. | AMS: Linear algebra and multiple algebra; proclaimed theory-Research presentation (book, total theory). MSC ｜ Ret trip geometry and discrete geometry-Research presentation (book / total theory). MSC ｜ Probability theory / probability process theory-Research presentation (book / total theory). MSC | Statistics-Scientific commentary (monograph, general theory). MSC ｜ Numerical Analysis-Scientific Commentary (Monograph, Review Article < SPAN> Life Mazseo, Opening Editor Michael V Mahoni, John K. Dochi, Anna K. Gilbert, Opening Editor IAS/ Park City Mathematics The Institute is a mathematics and mathematical education researcher, mathematics teachers, graduate students, and seniors, but this book is implemented in the mathematical education program that participates in research and education programs. Includes lectures on mathematical subjects in the summer school program 15-02, 60-02, 65-02, 68-02, 90-02, 68-02, 68-02, 90-02.

US Congress Library Catalog Inn publication Data Name Mahoney, Michael W., Editor. | Duchi, John, Editor. | | Publisher name: Mathematics Institute Park City. | Park City. Title: Data Mathematics / Michael V. MAHONI, John K. Duchi, Anna K. Gilbert, Editors. Park City Mathematics Series; Volume 25 | "Institute for Advanced Study "|" Industrial Mathematics Association "| | Includes a book magazine link. ID: lccn 2018024239 | ISBN 9781470435752 (Alk. Paper) Topics: LCSH: Mathematics-Teacher preparation-Conference. | Mathematics research and education conference | Big data conference. | AMS: Linear algebra and multiple algebra; proclaimed theory-Research presentation (book, total theory). MSC ｜ Ret trip geometry and discrete geometry-Research presentation (book / total theory). MSC ｜ Probability theory / probability process theory-Research presentation (book / total theory). MSC | Statistics-Scientific commentary (monograph, general theory). MSC ｜ Numerical Analysis-Scientific Commentary (Monograph, Review Article Life Mazseo, Maquel V. Mahoni, Michael V. Mahoni, John K. Dochi, Anna K. Gilbert, the opening editor, the Park City Mathematics Research Institute , Mathematics and mathematical education researchers, mathematics teachers, graduate students, and senior students are different, but this book is implemented in a mathematical education program in which they participate in research and education programs. Includes lecture notebooks for mathematics in school programs 15-02, 52-02, 65-02, 68-02, 90-02.

US Congress Library Catalog Inn publication Data Name Mahoney, Michael W., Editor. | Duchi, John, Editor. | | Publisher name: Mathematics Institute Park City. | Park City. Title: Data Mathematics / Michael V. MAHONI, John K. Duchi, Anna K. Gilbert, Editors. Park City Mathematics Series; Volume 25 | "Institute for Advanced Study "|" Industrial Mathematics Association "| | Includes a book magazine link. ID: lccn 2018024239 | ISBN 9781470435752 (Alk. Paper) Topics: LCSH: Mathematics-Teacher preparation-Conference. | Mathematics research and education conference | Big data conference. | AMS: Linear algebra and multiple algebra; proclaimed theory-Research presentation (book, total theory). MSC ｜ Ret trip geometry and discrete geometry-Research presentation (book / total theory). MSC ｜ Probability theory / probability process theory-Research presentation (book / total theory). MSC | Statistics-Scientific commentary (monograph, general theory). MSC ｜ Numerical Analysis-Scientific Commentary (Monograph, Review Article)

Copy and Reproduction and no n-profit libraries, which are specific readers of this magazine and their agency, can intentionally use the magazine materials, such as copying individual pages to use them for education or research activities. It is allowed to quote short sentences from this book to reviews on a regular basis for providing a link to the source regularly. Reproduction of materials from this book, organized copy, and multiple duplications are allowed only under the license of American Mathematic Society. Requests to resolve some reuse of AMS publications are processed by the Copyright Protection Center. For more information, see www. AMS. Org/Publications/Pubpermissions. Please send the transfer of rights and licensed separate printing to the address [ E-mail Protected]. All rights are protected. The US Mathematics Society retains all rights except for the US government. Printed in the United States. The paper used in this book does not contain acid and complies with the recommendation.

Established to ensure safety and sustainability. AMS homepage Please see HTTPS: // www.

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Lecture on randomization numeric lectures Petros DRINEAS and MICHAEL V. MAHONI

Optimization algorithm for data analysis

Introduction to probability optimization lecture John K. Dachi

Random method of matrix computing Peres-Gunnar Martinson

Four lectures on stochastic methods for data science Romantic Versinin

Homology algebra and data Robert Glast

IAS/Park City Mathematics Series 25, Page s-Page s-7-8 https: // doi. Org/10. 1090/PCMS/Pcms/025/00827 < SPAN> Copy and Reproduction This magazine's specific readers and no n-profit serving The library can intentionally use this magazine material, such as copying individual pages to use it for education or research activities. It is allowed to quote short sentences from this book to reviews on a regular basis for providing a link to the source regularly. Reproduction of materials from this book, organized copy, and multiple duplications are allowed only under the license of American Mathematic Society. Requests to resolve some reuse of AMS publications are processed by the Copyright Protection Center. For more information, see www. AMS. Org/Publications/Pubpermissions. Please send the transfer of rights and licensed separate printing to the address [ E-mail Protected]. All rights are protected. The US Mathematics Society retains all rights except for the US government. Printed in the United States. The paper used in this book does not contain acid and complies with the recommendation.

Established to ensure safety and sustainability. AMS homepage Please see HTTPS: // www.

23 22 21 20 19 18

Lecture on randomization numeric lectures Petros DRINEAS and MICHAEL V. MAHONI

Optimization algorithm for data analysis

Introduction to probability optimization lecture John K. Dachi

Random method of matrix computing Peres-Gunnar Martinson

US Congress Library Catalog Inn publication Data Name Mahoney, Michael W., Editor. | Duchi, John, Editor. | | Publisher name: Mathematics Institute Park City. | Park City. Title: Data Mathematics / Michael V. MAHONI, John K. Duchi, Anna K. Gilbert, Editors. Park City Mathematics Series; Volume 25 | "Institute for Advanced Study "|" Industrial Mathematics Association "| | Includes a book magazine link. ID: lccn 2018024239 | ISBN 9781470435752 (Alk. Paper) Topics: LCSH: Mathematics-Teacher preparation-Conference. | Mathematics research and education conference | Big data conference. | AMS: Linear algebra and multiple algebra; proclaimed theory-Research presentation (book, total theory). MSC ｜ Ret trip geometry and discrete geometry-Research presentation (book / total theory). MSC ｜ Probability theory / probability process theory-Research presentation (book / total theory). MSC | Statistics-Scientific commentary (monograph, general theory). MSC ｜ Numerical Analysis-Scientific Commentary (Monograph, Review Article Life Mazseo, Maquel V. Mahoni, Michael V. Mahoni, John K. Dochi, Anna K. Gilbert, the opening editor, the Park City Mathematics Research Institute , Mathematics and mathematical education researchers, mathematics teachers, graduate students, and senior students are different, but this book is implemented in a mathematical education program in which they participate in research and education programs. Includes lecture notebooks for mathematics in school programs 15-02, 52-02, 65-02, 68-02, 90-02.

Homology algebra and data Robert Glast

IAS/Park City Mathematics Series Volume 25, Page s-7-8 https: // doi. Org/10. 1090/PCMS/025/00827 Copy and Reproduction This is a specific reader and a no n-profitable library You can intentionally use the documents of this magazine, such as copying individual pages for education or research activities. It is allowed to quote short sentences from this book to reviews on a regular basis for providing a link to the source regularly. Reproduction of materials from this book, organized copy, and multiple duplications are allowed only under the license of American Mathematic Society. Requests to resolve some reuse of AMS publications are processed by the Copyright Protection Center. For more information, see www. AMS. Org/Publications/Pubpermissions. Please send the transfer of rights and licensed separate printing to the address [ E-mail Protected]. All rights are protected. The US Mathematics Society retains all rights except for the US government. Printed in the United States. The paper used in this book does not contain acid and complies with the recommendation.

Established to ensure safety and sustainability. AMS homepage Please see HTTPS: // www.

23 22 21 20 19 18

Lecture on randomization numeric lectures Petros DRINEAS and MICHAEL V. MAHONI

Optimization algorithm for data analysis

Introduction to probability optimization lecture John K. Dachi

Random method of matrix computing Peres-Gunnar Martinson

Four lectures on stochastic methods for data science Romantic Versinin

Homology algebra and data Robert Glast

23 22 21 20 19 18

The IAS/ Park City Mathematics Research Institute (PCMI) was established in 1991 as part of the National Geometric Research Institute of the National Science Fund. In mi d-1993, the program was established at Promise Research Institute (IAS) in Princeton, New Jersey. IAS/ Park City's Mathematics Institute encourages both mathematics research and education, contributing to the interaction between the two. The weekly research institute offers programs for researchers, graduate students, undergraduate students, high school students, undergraduate teachers, teachers from kindergartens to high school, and foreign teachers and researchers in educational fields. In the teacher instructor program, seminars and other events are held on weekends. One of the main purposes of PCMI is that all participants are familiar with all events held in the field of research, mathematical education, and mathematical education. We are trying to bring professional mathematicians in the educational process and attract teachers' interests from the modern concept of mathematics. For this purpose, in the afternoon during the summer training session, seminars and discussions will be held for all participants with a common interest and contributing to different groups. Many of them were established in 1991 as part of the Division's Geometric Research Institute of the National Science Fund in 1991. In mi d-1993, the program was established at Promise Research Institute (IAS) in Princeton, New Jersey. IAS/ Park City's Mathematics Institute encourages both mathematics research and education, contributing to the interaction between the two. The weekly research institute offers programs for researchers, graduate students, undergraduate students, high school students, undergraduate teachers, teachers from kindergartens to high school, and foreign teachers and researchers in educational fields. In the teacher instructor program, seminars and other events are held on weekends. One of the main purposes of PCMI is that all participants are familiar with all events held in the field of research, mathematical education, and mathematical education. We are trying to bring professional mathematicians in the educational process and attract teachers' interests from the modern concept of mathematics. For this purpose, in the afternoon during the summer training session, seminars and discussions will be held for all participants with a common interest and contributing to different groups. Many of the IAS/ Park City Mathematics Research Institute (PCMI) was established in 1991 as part of the Regional Geometric Research Institute of the National Science Fund. In mi d-1993, the program was established at Promise Research Institute (IAS) in Princeton, New Jersey. IAS/ Park City's Mathematics Institute encourages both mathematics research and education, contributing to the interaction between the two. The weekly research institute offers programs for researchers, graduate students, undergraduate students, high school students, undergraduate teachers, teachers from kindergarten to high school, and foreign teachers and researchers in educational fields. In the teacher instructor program, seminars and other events are held on weekends. One of the main purposes of PCMI is that all participants are familiar with all events held in the field of research, mathematical education, and mathematical education. We are trying to bring professional mathematicians in the educational process and attract teachers' interests from the modern concept of mathematics. For this purpose, in the afternoon during the summer training session, seminars and discussions will be held for all participants with a common interest and contributing to different groups. Many of them are

Random method of matrix computing Peres-Gunnar Martinson

Theory of computational problems (2000) Quantum field doctrine, supersymmetry and enumerative geometr y-Isomorphisms and their applications (2002) Geometric combinatorics (2004) Mathematical biology (2005) Low-dimensional (2006) Statistical mechanics (2007) Analytical and algebraic geometry (2008) Functions of Artinov L-functions (2009) Mathematics in image processing (2010) Modular fields in the Limann plane (2011) Geometric Doctrines of Groups (2012) Geometry Test (2013) Mathematics and Materials (2014) Modular Field Geometry and Representation Doctrines (2015)

American Mathematical Conversation publishes Student Mathematical Library materials for secondary schools and IAS/PCM I-Teacher-programmareks materials for teacher training programs from June to August. After more than 25 years, PCMI has maintained its spiritual vitality and continues to produce outstanding members from all areas of mathematics every year, from medal winners in each field to teachers in source secondary schools. Raif Mazzeo PCMI Director March 2017

IAS/Park City Mathematics Series Volume 25, Page s-9-12 https://doi. org/10. 1090/pcms/025/00828

Access "Data Mathematics" by Michael V. Mahoni, John K. Dachi, Anna K. Gilbert "Data Mathematics" -This is from June to August of Mathematical University Park City (PCMI) held in July 2016 It was the 26th daily content of the session. On our day, the latter is called "big data" or "data science" -Suddenly, for those who are more abstract and more than symbols in calculation or more applied fields. In general, it generally has the ability to increase this field. But one inference should spread this delusion. Eventually, for example, the data must be modeled by a matrix, a figure or a flat table, and if you perform such an operation with different types of data, it will be possible to wait, which is actually available. Is a specific mathematical structure that is forced to pretend, for example, linear algebra, graph and logic doctrine. The effectiveness of the operation on the data is that it depends not only on the perfect modeling of the data, but also on such a perfect simulated ignorance, roar, or fog. The editing itself can also be simulated, for example, to create these statements, such as whether editing is literally or approximated, or whether editing makes C. It is possible. < SPAN> Access "Data Mathematics" by Michael V Mahoni, John K. Dachi, Anna K. Gilbert "Data Mathematics" -This is from June to 8th of Mathematical University Park City (PCMI) held in July 2016. It was the 26th daily content of the session up to the month. On our day, the latter is called "big data" or "data science" -Suddenly, for those who are more abstract and more than symbols in calculation or more applied fields. In general, it generally has the ability to increase this field. But one inference should spread this delusion. Eventually, for example, the data must be modeled by a matrix, a figure or a flat table, and if you perform such an operation with different types of data, it will be possible to wait, which is actually available. Is a specific mathematical structure that is forced to pretend, for example, linear algebra, graph and logic doctrine. The effectiveness of the operation on the data is that it depends not only on the perfect modeling of the data, but also on such a perfect simulated ignorance, roar, or fog. The editing itself can also be simulated, for example, to create these statements, such as whether editing is literally or approximated, or whether editing makes C. It is possible. Access "Data Mathematics" by Michael V. Mahoni, John K. Dachi, Anna K. Gilbert "Data Mathematics" -This is from June to August of Mathematical University Park City (PCMI) held in July 2016 It was the 26th daily content of the session. On our day, the latter is called "big data" or "data science" -Suddenly, for those who are more abstract and more than symbols in calculation or more applied fields. In general, it generally has the ability to increase this field. But one inference should spread this delusion. Eventually, for example, the data must be modeled by a matrix, a figure or a flat table, and if you perform such an operation with different types of data, it will be possible to wait, which is actually available. Is a specific mathematical structure that is forced to pretend, for example, linear algebra, graph and logic doctrine. The effectiveness of the operation on data is that it depends not only on the perfect modeling of the data, but also on such a perfectly simulated ignorance, roaring sound, or fog. The editing itself can also be simulated, for example, to create these statements, such as whether editing is literally or approximated, or whether editing makes C. It is possible.

A wel l-known method for data modeling, for example, the M × N matrix is a natural method for describing M objects, and each of them is described by n symptoms. As a result of the algebra, the options that are even more difficult, these are the doctrine of active tests and linear operators, © 2018 South American Mathemative Conversation < Span> A wel l-known method for data modeling, for example, M × N procession. Is a natural way to describe M objects, each is described by n symptoms, and as a result of this linear algebra, more difficult options, these are active tests and linear operators. As a doctrine, © 2018 South American Mathematicl ConverSatio n-wel l-known methods for data modeling, for example, M × N-queue guarantees a natural method for describing M objects, and each of them is n. Described by individual symptoms, as a result of this linear algebra, more difficult options, these are the doctrine of active testing and linear operators, © 2018 South American MathematiclSection.

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As in the case of an accident introduction in the random numeric values algebra, there is no use of two synergy for stochastic appification. In this chapter, information on important convex experts will be provided, and the former gradient and subtra method will be explained to solve similar issues. These methods are generally considered to be ordinary methods, and are more likely to converge than 2 more advanced methods, which are close to established tasks such as Newton Law, but may be resistant to noise in optimization of the task itself. Is excellent. We will also examine the methods derived from the mirror, adaptive methods, and how to check the upper and lower world of these stochastic algorithms.

Michael V. Mahoni, John K. Dachi, Anna K. Gilbert < SPAN> Use two synergies for stochasticization, as in the case of an accident in the case of an accident with a random numeric value. Nothing. In this chapter, information on important convex experts will be provided, and the former gradient and subtra method will be explained to solve similar issues. These methods are generally considered to be ordinary methods, and are more likely to converge than 2 more advanced methods, which are close to established tasks such as Newton Law, but may be resistant to noise in optimization of the task itself. Is excellent. We will also examine the methods derived from the mirror, adaptive methods, and how to check the upper and lower world of these stochastic algorithms.

Michael V. Mahoni, John K. Dochi, Anna K. Gilbert Randominlation, as in the case of an accident introduction on the number of random numerical wards, do not use two synergies for stochasticization. 。 In this chapter, information on important convex experts will be provided, and the former gradient and subtra method will be explained to solve similar issues. These methods are generally considered to be ordinary methods, and are more likely to converge than 2 more advanced methods, which are close to established tasks such as Newton Law, but may be resistant to noise in optimization of the task itself. Is excellent. We will also examine the methods derived from the mirror, adaptive methods, and how to check the upper and lower world of these stochastic algorithms.

Michael V Mahoni, John K. Dachi, Anna K. Gilbert

In chapter 4, "Randomization methods for matrix computation", randomization methods for efficient computation of low-rank approximations of a given matrix are considered in more detail. It is often required to align a large matrix A of size M×N (M and N are large) with two rectangular matrices E and F of low rank such that a≈EF. Example problems are low-rank approximations of eigenvalue decompositions and decompositions along singular values. Low-rank acceptance problems of this kind are fundamental in traditional applied mathematics and scientific computing, but also arise in a wide range of data science applications. However, the key point is that the problems posed to these matrix collapses (e. g., when we are interested in numerical accuracy or statistical inference goals) and even the ways in which we access these matrices (e. g., as part of RAM model idealizations, or in single-passage streaming modes when we cannot even store the data) are quite different. Here, accidents are useful in many ways. In this chapter, we discuss randomization algorithms that allow us to exploit accidents to improve the communication properties of our algorithms, both in the RAM model and in the streaming model, and thus obtain the best possible working time in the worst case. In Chapter 4, "Randomization Methods for Matrix Computation", randomization methods for the efficient computation of low-rank approximations of a given matrix are considered in more detail. It is often required to align a large matrix A of size M×N (M and N are large) with two rectangular matrices E and F of low rank such that a≈EF. Example problems are low-rank approximations of eigenvalue decompositions and decompositions along singular values. Low-rank acceptance problems of this kind are fundamental in traditional applied mathematics and scientific computing, but also arise in a wide range of data science applications. However, the important thing is that the problems posed to these matrix collapses (e. g., when we are interested in numerical accuracy or statistical inference goals) and even the ways in which we access these matrices (e. g., as part of a RAM model idealization or in a single-passage streaming mode when we cannot even store the data) are quite different. Here, accidents are useful in many ways. In this chapter, we discuss randomization algorithms that can exploit accidents to improve the communication properties of algorithms, both in the RAM model and in the streaming model, and thus obtain the best possible working time in the worst case. In Chapter 4, "Randomization Methods for Matrix Computation", randomization methods for the efficient computation of low-rank approximations of a given matrix are considered in more detail. It is often required to align a large matrix A of size M×N (M and N are large) with two rectangular matrices E and F of low rank such that a≈EF. Example problems are low-rank approximations of eigenvalue decompositions and decompositions along singular values. Low-rank acceptance problems of this kind are fundamental in traditional applied mathematics and scientific computing, but also arise in a wide range of data science applications. However, the important thing is that the problems posed to these matrix collapses (e. g., when we are interested in numerical accuracy or statistical inference goals) and even the ways in which we access these matrices (e. g., as part of a RAM model idealization or in a single-passage streaming mode when we cannot even store the data) are quite different. Here, accidents are useful in many ways. In this chapter, we will discuss randomized algorithms, whether in the RAM model or the streaming model, that can take advantage of accidents to improve the communication properties of the algorithm, thereby obtaining the best possible working time in the worst case.

Random method of matrix computing Peres-Gunnar Martinson

The important point from the class of the data structure is that linear alternate analytics can be expanded not only in the 99, 9 % cases, but also to cover the order of linear conversion that forms complex numbers. It opens mathematical development opportunities. Overall, in the 2016 PCMI Summer Program, Petros Dorinius, John Dochi, Cynthia Palace, Kunar Talbala, Robert Grist, Peter Indian, Mauro Martinson, Gunner Martinson, Roman Versinin, Steven, Steven. A mini course with light was held. The book contains materials from Petros Dorinius (c o-authored with Michael Mahoni), Stephen Wright, John Douch, Gunner Martinson, Roman Versinine, and Robert Grista. Since each chapter of this volume was written by another author, each chapter has its own style, including the difference in the sect, but has been devised to make a fruitful reading. Collecting such works into one book is also a whole summer workshop, not easy, but it was not difficult for us to provide great support. First of all, I would like to thank the former PCMI software director Richard Hein, who first proposed the summer school, and the current PCMI software director Life Director Life Matzeo. The holding of the summer workshop is as follows.

1090/PCMS/025/01 IAS/Park City Mathematics Series Volume 25

23 22 21 20 19 18

1090/PCMS/025/01 IAS/Park City Mathematics Series Volume 25, Pages 1-48 https: // doi. Org/1090/PCMS/025/00829

Lecture on randomization numeric lectures on the dates, Petros Drinese and Michael v. MAHONI Data structure is important for the linear algebra is that linear algebra is not only for linear conversion generated in 99, 9 % cases, but also for linear conversion that forms complex numbers. The order can be expanded so that it can be covered, which opens further opportunities for mathematical development. Overall, in the 2016 PCMI Summer Program, Petros Dorinius, John Dochi, Cynthia Palace, Kunar Talbala, Robert Grist, Peter Indian, Mauro Martinson, Gunner Martinson, Roman Versinin, Steven, Steven. A mini course with light was held. The book contains materials from Petros Dorinius (c o-authored with Michael Mahoni), Stephen Wright, John Douch, Gunner Martinson, Roman Versinine, and Robert Grista. Since each chapter of this volume was written by another author, each chapter has its own style, including the difference in the sect, but has been devised to make a fruitful reading. Collecting such works into one book is also a whole summer workshop, not easy, but it was not difficult for us to provide great support. First of all, I would like to thank the former PCMI software director Richard Hein, who first proposed the summer school, and the current PCMI software director Life Director Life Matzeo. The holding of the summer workshop is as follows.

1090/PCMS/025/01 IAS/Park City Mathematics Series Volume 25, Pages 1-48 https: // doi. Org/1090/PCMS/025/00829

Lecture on randomization numeric lectures Petros DRINEAS and MICHAEL V. MAHONI

The importance of linear algebra 2. 2 2. 2) Land scales 2. 3 Generally recognized vectors. 2. 4 Measurements approved by the induction matrix. 2. 5 Frobenius Norm 2. 6 Disassembly in line with the 6 monetary price. 2. 7 SVD and basic matrix site. 2. 8 A universally recognized line shut. 2. 10 Links Discontinuation probability 3. 1 Random experiment: soil. 3. 2 Events. 3 Limit associations. 3. Autonomous acts and autonomous acts. 3. 5 Relative probability 3. 6 random numbers. 3. 7 Cumulative functions of major probability density function and dispersion function. 3. 8 Autonomous random numbers 3. 9 expected value of random number. 3. 10 Random values fluctuations 3. 11 4. 2 Test of methods for almost appropriate probability.

2 3 3 4 4 5 7 9 9 9 9 9 9 9 10 11 11 12 12 12 12 13 14 14 16 16 16 16 16 21

Arithmetic subject classification 2010. Home 68W20; Secondary 65FXX, 62Jxx. Main text and chirard random selection, random imaging, comparison of matrix, approximation of minimum square, approximation of lo w-next queue, mathematical university Park City school © 2018 Petros Dorinius, Michael V. Mahoni

Random method of matrix computing Peres-Gunnar Martinson

4. 3 The universal scale restrictions. 4. 4 Links 5. 1 Randomized reconstruction of Hadamado. 5. 2 Readers and protagonists. 5. 3 Preliminary RANDNL A-method. 5. 4 Confirmation of worship 5. 5 Working time for the RandleasTSQUARES method. 5. 6 RANDNLA method 6. 6. 2 Another formula 6. 3 structural inequality type 6. 3 for the approximation of the left low rank matrix. 6. 4 Confirmation of worth 6. 1. 1. 6. 5 Working time 6. 6 Left

21 24 24 24 25 26 26 31 36 36 36 37 41 47 47 < SPAN> Importance of linear algebra 2. 2. 2 2. 4 Measurements approved by the induction matrix. 2. 5 Frobenius Norm 2. 6 Disassembly in line with the 6 monetary price. 2. 7 SVD and basic matrix site. 2. 8 A universally recognized line shut. 2. 10 Links Discontinuation probability 3. 1 Random experiment: soil. 3. 2 Events. 3 Limit associations. 3. Autonomous acts and autonomous acts. 3. 5 Relative probability 3. 6 random numbers. 3. 7 Cumulative functions of major probability density function and dispersion function. 3. 8 Autonomous random numbers 3. 9 expected value of random number. 3. 10 Random values fluctuations 3. 11 4. 2 Test of methods for almost appropriate probability.

2 3 3 4 4 5 7 9 9 9 9 9 9 9 10 11 11 12 12 12 12 13 14 14 16 16 16 16 16 21

Arithmetic subject classification 2010. Home 68W20; Secondary 65FXX, 62Jxx. Main text and chirard random selection, random imaging, comparison of matrix, approximation of minimum square, approximation of lo w-next queue, mathematical university Park City school © 2018 Petros Dorinius, Michael V. Mahoni

Lecture on randomization numeric line algebra

4. 3 The universal scale restrictions. 4. 4 Links 5. 1 Randomized reconstruction of Hadamado. 5. 2 Readers and protagonists. 5. 3 Preliminary RANDNL A-method. 5. 4 Confirmation of worship 5. 5 Working time for the RandleasTSQUARES method. 5. 6 RANDNLA method 6. 6. 2 Another formula 6. 3 structural inequality type 6. 3 for the approximation of the left low rank matrix. 6. 4 Confirmation of worth 6. 1. 1. 6. 5 Working time 6. 6 Left

23 22 21 20 19 18

2 3 3 4 4 5 7 9 9 9 9 9 9 9 10 11 11 12 12 12 12 13 14 14 16 16 16 16 16 21

Arithmetic subject classification 2010. Home 68W20; Secondary 65FXX, 62Jxx. Main text and chirard random selection, random imaging, comparison of matrix, approximation of minimum square, approximation of lo w-next queue, mathematical university Park City school © 2018 Petros Dorinius, Michael V. Mahoni

Lecture on randomization numeric line algebra

4. 3 The universal scale restrictions. 4. 4 Links 5. 1 Randomized reconstruction of Hadamad. 5. 2 Readers and protagonists. 5. 3 Preliminary RANDNL A-method. 5. 4 Confirmation of worth 5. 5 Working time for the RandleasTSQUARES method. 5. 6 RANDNLA method 6. 6. 2 Another formula 6. 3 structural inequality type 6. 3 for the approximation of the left low rank matrix. 6. 4 Confirmation of worth 6. 1. 1. 6. 5 Working time 6. 6 Left

21 24 24 24 25 26 28 31 35 36 36 37 40 41 47 47 47

1. The matrix input is widely used in computer science, statistics, and application arithmetic. The M × N matrix has the ability to sign information about M objects (each written in n characters) and the behavioral behavior of the sampled differential operators on a lattice of a limited element. The n x N × N has the ability to sign the correlation between all pairs of n objects, such as public networks and all components. In recent years, thanks to the development of technology that generates very huge scientific data and Internet data, the doctrine and practice of queue algorithms have been attractive in many ways. Furthermore, randomization (usually, data generation (usually data generation) as an algorithm and calculation resource for developing improved algorithms such as tasks, passing lines, minimum square (LS), approximation of the minimum square (LS), and approximation of lo w-ranked processions. It is worth notable that the introduction of the input data characteristics for the noise in the mechanism of the mechanism. Randomized numeric line algebra (RANDNLA) is an interdisciplinary research field that randomized as computing sources for developing improved algorithm for larg e-scale tasks of linear algebra. From the basics

Petros Dorinius in Michael V. Mahoni is widely used in computer science, statistics, and application arithmetic. The M × N matrix has the ability to sign information about M objects (each written in n characters) and the behavioral behavior of the sampled differential operators on a lattice of a limited element. The n x N × N has the ability to sign the correlation between all pairs of n objects, such as public networks and all components. In recent years, thanks to the development of technology that generates very huge scientific data and Internet data, the doctrine and practice of queue algorithms have been attractive in many ways. Furthermore, randomization (usually, data generation (usually data generation) as an algorithm and calculation resource for developing improved algorithms such as tasks, passing lines, minimum square (LS), approximation of the minimum square (LS), and approximation of lo w-ranked processions. It is worth notable that the introduction of the input data characteristics for the noise in the mechanism of the mechanism. Randomized numeric line algebra (RANDNLA) is an interdisciplinary research field that randomized as computing sources for developing improved algorithm for larg e-scale tasks of linear algebra. From the basics

Petros Dorinius 1. in Michael V. Mahoni is widely used in computer science, statistics, and application arithmetic. The M × N matrix has the ability to sign information about M objects (each written in n characters) and the behavioral behavior of the sampled differential operators on a lattice of a limited element. The n x N × N has the ability to sign the correlation between all pairs of n objects, such as public networks and all components. In recent years, thanks to the development of technology that generates very huge scientific data and Internet data, the doctrine and practice of queue algorithms have been attractive in many ways. Furthermore, randomization (usually, data generation (usually data generation) as an algorithm and calculation resource for developing improved algorithms such as tasks, passing lines, minimum square (LS), approximation of the minimum square (LS), and approximation of lo w-ranked processions. It is worth notable that the introduction of the input data characteristics for the noise in the mechanism of the mechanism. Randomized numeric line algebra (RANDNLA) is an interdisciplinary research field that randomized as computing sources for developing improved algorithm for larg e-scale tasks of linear algebra. From the basics

Random method of matrix computing Peres-Gunnar Martinson

Exhibits excellent scalability in parallel media and distributed media for these common tasks, for example, regression with the minimum square method. More than that, Randnla promises powerful algorithms and statistical basics for advanced larg e-scale data analysis. This leader functions as a sel f-fulfilled and smooth access to three basic RANDNLA methods (randomization of matrix randomization, randomization method of minimum square methods, randomization method that calculates lo w-ranking approximation of queue). 。 Thus, the leader has a close connection with almost all the arithmetic techniques, especially with some other chapters in this volume. First, G. Martinson research, R. Vershinin research [28], and J. Duchi research [12] It is. In this chapter, the first examples of the linear algebra are first examined in the two verse, and in the three verse, the preceding case of the discrete probability is considered, the randomization method is given in verse 4, and the smallest tw o-square is returned in verse 5. The randomization method for tasks is given, and finally a randomization method for approximation of low rank is given in verse six. In this introduction, [10, 17] research still demonstrates excellent scalability in parallel media and distributed media, in response to R < Span> general tasks, for example, regression by the minimum square method. More than that, Randnla promises powerful algorithms and statistical basics for advanced larg e-scale data analysis. This leader functions as a sel f-fulfilled and smooth access to three basic RANDNLA methods (randomization of matrix randomization, randomization method of minimum square methods, randomization method that calculates lo w-ranking approximation of queue). 。 Thus, the leader has a close connection with almost all the arithmetic techniques, especially with some other chapters in this volume. First, G. Martinson research, R. Vershinin research [28], and J. Duchi research [12] It is. In this chapter, the first examples of the linear algebra are first examined in the two verse, and in the three verse, the preceding case of the discrete probability is considered, the randomization method is given in verse 4, and the smallest tw o-square is returned in verse 5. The randomization method for tasks is given, and finally a randomization method for approximation of low rank is given in verse six. In this introduction, [10, 17] still demonstrates excellent scalability in parallel media and distributed media, for these general tasks, for example, regression by the minimum square method. More than that, Randnla promises powerful algorithms and statistical basics for advanced larg e-scale data analysis. This leader functions as a sel f-fulfilled and smooth access to three basic RANDNLA methods (randomization of matrix randomization, randomization method of minimum square methods, randomization method that calculates lo w-ranking approximation of queue). 。 Thus, the leader has a close connection with almost all the arithmetic techniques, especially with some other chapters in this volume. First, G. Martinson research, R. Vershinin research [28], and J. Duchi research [12] It is. In this chapter, the first examples of the linear algebra are first examined in the two verse, and in the three verse, the preceding case of the discrete probability is considered, the randomization method is given in verse 4, and the smallest tw o-square is returned in verse 5. The randomization method for tasks is given, and finally a randomization method for approximation of low rank is given in verse six. In this introduction, the research of [10, 17] is still R.

2. In the line algebra, starts with a simple educational program called the main preceding example of the linear algebra used in this chapter. Basic knowledge of linear algebra (for example, internal/ external operations of vector, main operations of queue, queue as the main body structure, multiplication of scalar, upper/ lower triangle matrix, queue-multiplication, multiplication I would be happy if there was a matrix printing. 2. 1. Below. We concentrate on the queue and vector to the amount of material. We start applying the name of X∈RN to specify the N-MURGED vector: be careful when introducing the vector Latin lowercase. If the vector is not clearly stipulated, the vector will be a vector-player every time. All zero vectors are classified as 0, and all units vectors are classified as 1; the dimension is assumed from the context or the appearance of the replacement-index. To represent a line, we begin to apply thick Latin lowercase. For example, A ∈ RM x N means the volume M × n of the matrix A, to start applying AI * to represent the I, row A, as a vector of the row, and represent the II column A as a column vector. Start applying A * i to. The square queue of the equity formula is classified as follows.

Randomized numerical lin e-up algebra < Span> 2. Linear algebra is started with a simple educational program called a major precedent for linear algebra used in this chapter. Basic knowledge of linear algebra (for example, internal/ external operations of vector, main operations of queue, queue as the main body structure, multiplication of scalar, upper/ lower triangle matrix, queue-multiplication, multiplication I would be happy if there was a matrix printing. 2. 1. Below. We concentrate on the queue and vector to the amount of material. We start applying the name of X∈RN to specify the N-MURGED vector: be careful when introducing the vector Latin lowercase. If the vector is not clearly stipulated, the vector will be a vector-player every time. All zero vectors are classified as 0, and all units vectors are classified as 1; the dimension is assumed from the context or the appearance of the replacement-index. To represent a line, we begin to apply thick Latin lowercase. For example, A ∈ RM x N means the volume M × n of the matrix A, to start applying AI * to represent the I, row A, as a vector of the row, and represent the II column A as a column vector. Start applying A * i to. The square queue of the equity formula is classified as follows.

Randomized numerical line type algebra 2. In the lin e-shaped algebra, it starts with a simple educational program called the main precedent of linear algebra, which is useful in this chapter. Basic knowledge of linear algebra (for example, internal/ external operations of vector, main operations of queue, queue as the main body structure, multiplication of scalar, upper/ lower triangle matrix, queue-multiplication, multiplication I would be happy if there was a matrix printing. 2. 1. Below. We concentrate on the queue and vector to the amount of material. We start applying the name of X∈RN to specify the N-MURGED vector: be careful when introducing the vector Latin lowercase. If the vector is not clearly stipulated, the vector will be a vector-player every time. All zero vectors are classified as 0, and all units vectors are classified as 1; the dimension is assumed from the context or the appearance of the replacement-index. To represent a line, we begin to apply thick Latin lowercase. For example, A ∈ RM x N means the volume M × n of the matrix A, to start applying AI * to represent the I, row A, as a vector of the row, and represent the II column A as a column vector. Start applying A * i to. The square queue of the equity formula is classified as follows.

23 22 21 20 19 18

Revolutionary matrix. The matrix a∈Rn × N is considered non-hector or reversal if a matrix A-1∈Rn × n is present such as AAA-1 = in × n = a-1 A. Other texts are not actually unnecessary vectors X∈RN like AX = 0. The normal qualities of the reversal are as follows: ( a-1) = (a) -1 = a- and (ab) -1 = b-1 a-1. Orthogonal procession. The matrix a∈Rn × N is an orthogonal matrix if A = A-1. This is equivalent to 0 for all I and j from 1 to N, when I = j a ∗ i a ∗ j =. 1. In the case of I = j, the same property is correct, and in other texts, the pair of columns (rows) is orthodox, regarded as a normal vector. Q R-outfit. In any matrix a∈RN x n, the following may be laid out in the orthogonal and upper triangular matrix: here, Q (RN x N) is a orthogonal matrix, r (R ( RN x N) is a triangular matrix. The QR output is excellent in solving the collective primary equation, and the calculation amount is O (n3), and is considered to be numerically permanent. In order to solve the collective primary equation type AX = B that supports Q R-Extension, QRX = RX = Q B is first applied to both sides. 2. 2. The generalized scale that is generally recognized is used

Petros Dorinius, Michael V. Mahoni

The vector P Norm is the most widely thought: -S single norm: x1 = n i = 1 | √ n 2-Euclid (twin vad) Norm: x2 = i = 1 | = x x. -Long (maximum) Norm: In XWORK = Max1in | XI |. X, y∈Rn, we can bind the internal work x y = generally recognized measured values. Koshi-shvartz Inequal formula describes:

Random method of matrix computing Peres-Gunnar Martinson

Petros Dorinius, Michael V. Mahoni

The vector P Norm is the most widely thought: -S single norm: x1 = n i = 1 | √ n 2-Euclid (twin vad) Norm: x2 = i = 1 | = x x. -Long (maximum) Norm: In XWORK = Max1in | XI |. X, y∈Rn, we can bind the internal work x y = generally recognized measured values. Koshi-shvartz Inequal formula describes:

| X Y ｜ X2 Y2. Other texts are distinguished from the two vector data of the internal work of the two vectors in the definition of the universally recognized data. Gelder's inequality is described as follows. Revolutionary matrix. The matrix a∈Rn × N is considered non-hector or reversal if a matrix A-1∈Rn × n is present such as AAA-1 = in × n = a-1 A. Other texts are not actually unnecessary vectors X∈RN like AX = 0. The normal qualities of the reversal are as follows: ( a-1) = (a) -1 = a- and (ab) -1 = b-1 a-1. Orthogonal procession. The matrix a∈Rn × N is an orthogonal matrix if A = A-1. This is equivalent to 0 for all I and j from 1 to N, when I = j a ∗ i a ∗ j =. 1. In the case of I = j, the same property is correct, and in other texts, the pair of columns (rows) is orthodox, regarded as a normal vector. Q R-outfit. In any matrix a∈RN x n, the following may be laid out in the orthogonal and upper triangular matrix: here, Q (RN x N) is a orthogonal matrix, r (R ( RN x N) is a triangular matrix. The QR output is excellent in solving the collective primary equation, and the calculation amount is O (n3), and is considered to be numerically permanent. In order to solve the collective primary equation type AX = B that supports Q R-Extension, QRX = RX = Q B is first applied to both sides. 2. 2. The generalized scale that is generally recognized is used

Petros Dorinius, Michael V. Mahoni

23 22 21 20 19 18

| X Y ｜ X2 Y2. Other texts are distinguished from the two vector data of the internal work of the two vectors in the definition of the universally recognized data. Gelder's inequality is described as follows.

Simply split the right inequality between the vector P-Norm: XIP x1 NX et, √ x2 X1 NX2, √ xy x2 NX et. Further X22 = XT X. You can configure the Pythagoras axiom. Priority 2. 2. 2 Vector x, y∈Rn is orthogonal, that is, x y = 0. Axiomes 2. 1 is still famous as Pythagoras axiom. Another interesting nature of Euclid's universal scale is that it does not change (after) before multiplication (lines) with orthogonal columns (lines). Prix 2. 3. 2. Vector X∈Rn and matrix v∈rm × n for mn and v = in: VX2 = x2 and XT V t 2 = x2 are set. 2. 4. Generalization of induction matrix. Considering the matrix a∈rm x n and one amount p 1, determine the p-norm in the queue as follows: AXP = Max AXP. AP = Max x = 0 XP XP = 1 The P-Norm is as follows: -1 Norm: maximum unconditional sum of columns, MA1 = Max | = Max AEJ 1. 1jn

-Longnorm: Maximum unconditional sum of rows, AOWN = MAX

Random method of matrix computing Peres-Gunnar Martinson

Randomization numeric line algebraic lecture

- 2-dimensional (or spectral) Norm: A2 = Max AX2 = Max X2 = 1

In this way, there is a vector X of a universal measurement (P-Norm's single norm) similar to AP = AXP. The p-norm of the induced matrix follows the low-value law: AXP AP XP < SPAN> The right inequality between the joint P-norm of the vector: XIP x1 NX et, √ x2 X1 NX2, √ xy x2 nx et. Further x22 = xt x. Here, the concept of orthogonal vector can be modified and the pythagoras can be configured. Priority 2. 2. 2 Vector x, y∈Rn is orthogonal, that is, x y = 0. Axiomes 2. 1 is still famous as Pythagoras axiom. Another interesting nature of Euclid's universal scale is that it does not change (after) before multiplication (lines) with orthogonal columns (lines). Prix 2. 3. 2. Vector X∈Rn and matrix v∈rm × n for mn and v = in: VX2 = x2 and XT V t 2 = x2 are set. 2. 4. Generalization of induction matrix. Considering the matrix a∈rm x n and one amount p 1, determine the p-norm in the queue as follows: AXP = Max AXP. AP = Max x = 0 XP XP = 1 The P-Norm is as follows: -1 Norm: maximum unconditional sum of columns, MA1 = Max | = Max AEJ 1. 1jn

-Longnorm: Maximum unconditional sum of rows, AOWN = MAX

| AIJ | = Max A EI 1. 1JM

Randomization numeric line algebraic lecture

- 2-dimensional (or spectral) Norm: A2 = Max AX2 = Max X2 = 1

Random method of matrix computing Peres-Gunnar Martinson

-Longnorm: Maximum unconditional sum of rows, AOWN = MAX

| AIJ | = Max A EI 1. 1JM

Randomization numeric line algebraic lecture

- 2-dimensional (or spectral) Norm: A2 = Max AX2 = Max X2 = 1

In this way, there is a vector X of a universal measurement (P-Norm's single norm) similar to AP = AXP. The p-norm of the induced matrix follows the low-value law: AXP AP XP

Apart from this, the P-quota matrix is immutable to a line: P and Q are the correct dimensions. Similarly, if you look at a matrix with a double broken line and a column B A12, it becomes PAQ = A21 A22, and the ratio of submotones is associated with the absolutely no n-permanent queue Norm: BP AP. Justifying the correct relationship between the matrix P-Norm is relatively easy. In the case of matrix a∈rm x n, √ 1 √ a2 ma, n √ 1 √ A1 A1 A2 na1. M includes a 1 = aux and at least = a1. The transfer affects the universal universal matrix, but does not affect the double norm, that is, A2 = A2. Public (or follo w-up) of the column (or row) is considered an orthogonal vector: UAV 2 = a2, U and V are the corresponding dimensional orthogonal matrix (UT U = i and V Tv = I). This does not affect. 2. 5. Flobenius Norm. Flobenius Norm is not considered as a guided norm, which belongs to the universally recognized shut tribe (considered in verse 2. 8). 2. 5. Details 1. The matrix A∈rm x n determines the Norm of Flobenius as follows: M

N AF = A2IJ = Tr A, J = 1 i = 1

23 22 21 20 19 18

Petros Dorinias and Michael V. Mahoni < Span>, apart from this, the P-quota matrix is immutable to a line: P and Q are the right dimensions. Similarly, if you look at a matrix with a double broken line and a column B A12, it becomes PAQ = A21 A22, and the ratio of submotones is associated with the absolutely no n-permanent queue Norm: BP AP. Justifying the correct relationship between the matrix P-Norm is relatively easy. In the case of matrix a∈rm x n, √ 1 √ a2 ma, n √ 1 √ A1 A1 A2 na1. M includes a 1 = aux and at least = a1. The transfer affects the universal universal matrix, but does not affect the double norm, that is, A2 = A2. Public (or follo w-up) of the column (or row) is considered an orthogonal vector: UAV 2 = a2, U and V are the corresponding dimensional orthogonal matrix (UT U = i and V Tv = I). This does not affect. 2. 5. Flobenius Norm. Flobenius Norm is not considered as a guided norm, which belongs to the universally recognized shut tribe (considered in verse 2. 8). 2. 5. Details 1. The matrix A∈rm x n determines the Norm of Flobenius as follows: M

N AF = A2IJ = Tr A, J = 1 i = 1

Here, Tr (-) means the imprint of the matrix (reminiscent that the imprint in the square is arranged as a sum of the main components above the main diagonal). Unofficially, Flobenius Norm determines the distribution or variable of the matrix (which can be interpreted as a mass or mass). In the case of vector x∈RN, Flobenius Norm is equal to its Euclid Norm, that is, XF = X2. The transfer of the matrix a∈rm x n does not affect the Norm of Flobenius.

Apart from Petros Dorinius and Michael V. Mahoni, the P-quota matrix is unchanged against a line: P-and Q are the right dimensions. Similarly, if you look at a matrix with a double broken line and a column B A12, it becomes PAQ = A21 A22, and the ratio of submotones is associated with the absolutely no n-permanent queue Norm: BP AP. Justifying the correct relationship between the matrix P-Norm is relatively easy. In the case of matrix a∈rm x n, √ 1 √ a2 ma, n √ 1 √ A1 A1 A2 na1. M includes a 1 = aux and at least = a1. The transfer affects the universal universal matrix, but does not affect the double norm, that is, A2 = A2. Public (or follo w-up) of the column (or row) is considered an orthogonal vector: UAV 2 = a2, U and V are the corresponding dimensional orthogonal matrix (UT U = i and V Tv = I). This does not affect. 2. 5. Flobenius Norm. Flobenius Norm is not considered as a guided norm, which belongs to the universally recognized shut tribe (considered in verse 2. 8). 2. 5. Details 1. The matrix A∈rm x n determines the Norm of Flobenius as follows: M

Random method of matrix computing Peres-Gunnar Martinson

Here, Tr (-) means the imprint of the matrix (reminiscent that the imprint in the square is arranged as a sum of the main components above the main diagonal). Unofficially, Flobenius Norm determines the distribution or variable of the matrix (which can be interpreted as a mass or mass). In the case of vector x∈RN, Flobenius Norm is equal to its Euclid Norm, that is, XF = X2. The transfer of the matrix a∈rm x n does not affect the Norm of Flobenius.

Petros Dorinius and Michael V. Mahoni

Or, (after) -ulosice in front of each matrita with a regular orthetic row (after): U and V are the corresponding dimensional orthogonal matrix (UT U = I and V Tv = I). Flobenius Due and Norm are as follows: A2 AF Rank (a) A2 minus A2. Flobenius Norm, that is, to meet the strong lower pulling SO-DRIVEN characteristics: ABF A2 BF and ABF AF B2. For all vector x∈rm and y∈RN, their outer stacks are equivalent to the two vector Euclid Norms, which form the outer stack: Xy F = X2 Y2. Finally, a queue of the Pythagoras theorem is settled. Theorem 2. 5. 2 (matrecraft Pythagoras). A, b∈rm x n. If b = 0, A + B2F = a2f + B2F. 2. 6. Decomposition specific value decomposition (SVD) of peculiar values (SVD) is a major matrix decomposition and exists in any matrix. 2. 6. Determination of peculiar values 1. When a matrix a∈rm x n, the complete SVD is defined as follows: minimum

Here, u∈rm x m and v∈Rn x n are orthogonal in orthogonal pythema containing a peculiar vector A. A column U (each V) is represented using m (J = 1,., N). Similarly, σi, i = 1, ... Secal value A is not losing, and the number is equal to min. The number of unique values A that is not equal is equivalent to rank A. Due to an orthogonal virtue, it is as follows: σPaqt = σa, and here p and q are the corresponding orthogonal matrix (P = I, Qt Q = I). In other words, the peculiar value of PAQ matches the specific value A.

Randomization numeric line algebra

I = 1, ... min, | σi (a) - σi (b) | a-b2.

Next, at the time of a∈RP x m and b∈rm x n, all I = 1, ..., σi (ab) σ1 (a) σi (b), < SPAN> With (lines) in front of each matrita- (later) -ulosice: U and V are the corresponding dimensional orthogonal matrix (UT U = i and V T V = I). Flobenius Due and Norm are as follows: A2 AF Rank (a) A2 minus A2. Flobenius Norm, that is, to meet the strong lower pulling SO-DRIVEN characteristics: ABF A2 BF and ABF AF B2. For all vector x∈rm and y∈RN, their outer stacks are equivalent to the two vector Euclid Norms, which form the outer stack: Xy F = X2 Y2. Finally, a queue of the Pythagoras theorem is settled. Theorem 2. 5. 2 (matrecraft Pythagoras). A, b∈rm x n. If b = 0, A + B2F = a2f + B2F. 2. 6. Decomposition specific value decomposition (SVD) of peculiar values (SVD) is a major matrix decomposition and exists in any matrix. 2. 6. Determination of peculiar values 1. When a matrix a∈rm x n, the complete SVD is defined as follows: minimum

Here, u∈rm x m and v∈Rn x n are orthogonal in orthogonal pythema containing a peculiar vector A. The column U (each V) is represented using m (J = 1,., N). Similarly, σi, i = 1, ... Secal value A is not losing, and the number is equal to min. The number of unique values A that is not equal is equivalent to rank A. Due to an orthogonal virtue, it is as follows: σPaqt = σa, and here p and q are the corresponding orthogonal matrix (P = I, Qt Q = I). In other words, the peculiar value of PAQ matches the specific value A.

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I = 1, ... min, | σi (a) - σi (b) | a-b2.

Next, at the time of a∈RP x m and b∈rm x n, all I = 1, ..., min, σi (ab) σ1 (a) σi (b), or regular orthodox column (line) (Later) wit h-ulosice with each matrita with each matrita: U and V are the corresponding dimensional orthogonal matrix (UT U = i and V Tv = I). Flobenius Due and Norm are as follows: A2 AF Rank (a) A2 minus A2. Flobenius Norm, that is, to meet the strong lower pulling SO-DRIVEN characteristics: ABF A2 BF and ABF AF B2. For all vector x∈rm and y∈RN, their outer stacks are equivalent to the two vector Euclid Norms, which form the outer stack: Xy F = X2 Y2. Finally, a queue of the Pythagoras theorem is settled. Theorem 2. 5. 2 (matrecraft Pythagoras). A, b∈rm x n. If b = 0, A + B2F = a2f + B2F. 2. 6. Decomposition specific value decomposition (SVD) of peculiar values (SVD) is a major matrix decomposition and exists in any matrix. 2. 6. Determination of peculiar values 1. When a matrix a∈rm x n, the complete SVD is defined as follows: minimum

Here, u∈rm x m and v∈Rn x n are orthogonal in orthogonal pythema containing a peculiar vector A. The column U (each V) is represented using m (J = 1,., N). Similarly, σi, i = 1, ... Secal value A is not losing, and the number is equal to min. The number of unique values A that is not equal is equivalent to rank A. Due to an orthogonal virtue, it is as follows: σPaqt = σa, and here p and q are the corresponding orthogonal matrix (P = I, Qt Q = I). In other words, the peculiar value of PAQ matches the specific value A.

Randomization numeric line algebra

Random method of matrix computing Peres-Gunnar Martinson

Next, when A∈RP x m and b∈rm x n, all I = 1, ..., σi (ab) σ1 (a) σi (b),

Here, I remember σ1 (a) = a2. In the case of a rank (a) = ρ in a matrix a∈rm x n, the elegant SVD can be modified with URBAN. Definition 2. 4. Standing A∈rm × N rank ρ We decide a delicate SVD: ρ σ v = σi ui v = i = = = = = = = = ： ： ： ：

Here, u∈ and v∈ are a matrix with a pair of regular orthracks (that is, ut u = i and v tv = i), and is a discrete and honest spectrum vector A that is correctly unnecessary in general values. There is σ∈R × ρ σ × × ρ is a diagonal matrix having an unlimited equivalent value a in the order of the diagonal decrease. RN x ρ

If A is a non-allowable matrix, a revolutionary value can be determined by SVD: a-1 = (ulyv) -1 = VLARUS-1 u. (if A is incomparable, it is a square And it is a complete number, and in this case, the elegant SVD matches the absolute SVD). For example, apart from SVD, SVD is a basic because you can determine the optimal K-attachment for each matrix. Priority 2 6. 5. 5. A = Ulyv ∈ RM x N-Elegant SV D-MATRICES a v k. During this time, σk+1 = and And

A-b2f = a-AK 2F.

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Petros Dorinius, Michael V. Mahoni

Lower ρ-K's non-equal valid (right) peculiar vector A matrix; and σk, ⊥ ⊥ R (ρ-k) × (ρ-k) × 角 角 角 角 対 対 対 対 対 対 対 対 対 対 対;;; Is to represent.

And ak, ⊥ = a-ak = uk, ⊥ σk, ⊥ v tk, ⊥.

2. 7. SVD and basic matrix space. Each matrix A∈rm × N defines four basic locations: column A location: This location is covered by column A: range (A) =.

Location A: This place involves all vector x∈Rn, actually AX = 0: Null (A) = ⊂ Rn. ) =.

Null location on the left A: This place clips all vector y∈rm, and in fact, Ay = 0: null (a) = ⊂ rm. SVD identifies the orthogonal bases of all these places. If the matrix a∈rm × n is set and rank (a) = ρ, the SVD can be included in the appearance: σ v ρ 0 ρ. A = uρ u, ⊥ v 0 0 ρ, just just right: range (a) = range (uρ), null (a) = range (v ρ, ⊥), range (a) = range (v ρ), null (a) = range (uρ, ⊥). The position of the row of the orthogonal nurse space A and its group are RM. The trajectory of the column is a orthogonal zero space A, and those groups are RN. 2. 8. A universal matrix matrix matrix. The universally recognized shutten-major in the line is a special type of major recognized major recognized vectors, including the unique meaning of the matrix. Specific value σ1 - σ & amp; amp; AMP; AMP; A ∈ RM × n is given; 0, we recognize the p norma shuttle as follows: 1 ρ p P σi. AP = I = 1

And ak, ⊥ = a-ak = uk, ⊥ σk, ⊥ v tk, ⊥.

Random numerical line algebra

SHUTTEN 1 member Norm: Nuclear Norm, that is, a peculiar value. Shutten 2D Norm: Flobenius Norm, that is, a square of square of tw o-sided value. Shutten Norm Length: Shuttle's Side P → φ P-Norm, the spectrum defined, the largest specific value. The universal recognized shutten measurement is an orthogonal, unchanged and sophisticated, and satisfies the gelter inequality. 2. 9. Pseud o-flip of Mura Penroose. The generalization of general opinions regarding the reverse matrix of the matrix is the Mura Penroose pseud o-reverse matrix. When a matrix a∈rm × n is given, the matrix A † is deemed to be a Mura Penroose A pseud o-flip if it satisfies the following properties: (1) (2) (3) (4).

aa † a = a. aa † aa † = a †. (aa †) = aa †.

Standing A∈rm x N rank ρ and its elegant SVD A = RM x N rank ρ and its elegant SVD A = RM x N rank

Its pseud o-inverse Mura Penroose

This is ρ 1 vi u. Σi I = 1.

If A † = a-1. If a † = a-1. A∈rm × n contains the absolute column ranking, the ultimate row of A † a = in and aa † -column A; absolute small ranking If included, the projecting line to AA † = IM and A † A line A. [5, axim 2.

Random method of matrix computing Peres-Gunnar Martinson

(The job in the reverse matrix of the two queues is always equal to the job of the reverse matrix in these queues, but the same thing is not a complete sense in Mura Penrose [5].) The basic space of Mura Penroose's pseud o-reversal is associated with the real queue space. When the matrix A and its pseud o-reversed Mura Penroose A † are set, the column a † space can be determined as follows: Range (a †) = Range (a), Range (a),

Petros Dorinius and Michael V. Mahoni

And it is orthogonal to the null space A. Null space A † can be defined as follows: null (A †) = null (AA) = null (A), and it is orth orth orthetics of column A. Left. See [5, 13, 26, 27] for additional information about linear algebra and matrix calculation, and [4, 25] for additional information on the theory of matrix intake. < SPAN> (The job in the reverse matrix in the two matrix is always equal to the job of the inverted matrix in these queues, but the same thing is not a complete sense in Mura Penrose [5]. 。 The basic space of Mura Penroose's pseud o-reversal is associated with the real queue space. When the matrix A and its pseud o-reversed Mura Penroose A † are set, the column a † space can be determined as follows: Range (a †) = Range (a), Range (a),

Petros Dorinius and Michael V. Mahoni

And it is orthogonal to the null space A. Null space A † can be defined as follows: null (A †) = null (AA) = null (A), and it is orth orth orthetics of column A. Left. See [5, 13, 26, 27] for additional information about linear algebra and matrix calculation, and [4, 25] for additional information on the theory of matrix intake. (The job in the reverse matrix of the two queues is always equal to the job of the reverse matrix in these queues, but the same thing is not a complete sense in Mura Penrose [5].) The basic space of Mura Penroose's pseud o-reversal is associated with the real queue space. When the matrix A and its pseud o-reversed Mura Penroose A † are set, the column a † space can be determined as follows: Range (a †) = Range (a), Range (a),

Petros Dorinius and Michael V. Mahoni

23 22 21 20 19 18

3. Discrete probability This section adopts an educational program with a simple discrete probability. A more difficult result (especially inequality in the similarity of Bernstein to the probability of the substance and the probability of the matrix) is shown in the later context of this chapter. It is worth noting that Landra's huge share (not continuous probability) is based on a simple and basic principle of separation probability. 3. 1. Random experiment: Lower random experiences are all steps, including a specific set of stochastic results, with infinity. Bone play and coin throwing are common examples. The trajectory of random experience is a large number of stochastic results of random experience. This is often referred to as a random experience (for example, victory and failure). In the doctrine of discrete probability, the location of the sample ω is considered finite. (In this chapter, we do not consider the place of samples or countless places.) The event is an arbitrary part of the Sample Ω location. Definitely, many of the probable events are considered to be a gathering (many probable stations)

If E is an event, (3. 1. 1)

That is, the possibility of an act is equal to the possibility of its components. This departure follows the footprints of the facts of the possibility of empty action (corresponding to event E,).

Randomized numerical line algebra < Span> 3. Discrete probability This section adopts an educational program with a simple discrete probability. A more difficult result (especially inequality in the similarity of Bernstein to the probability of the substance and the probability of the matrix) is shown in the later context of this chapter. It is worth noting that Landra's huge share (not continuous probability) is based on a simple and basic principle of separation probability. 3. 1. Random experiment: Lower random experiences are all steps, including a specific set of stochastic results, with infinity. Bone play and coin throwing are common examples. The trajectory of random experience is a large number of stochastic results of random experience. This is often referred to as a random experience (for example, victory and failure). In the doctrine of discrete probability, the location of the sample ω is considered finite. (In this chapter, we do not consider the place of samples or countless places.) The event is an arbitrary part of the Sample Ω location. Definitely, many of the probable events are considered to be a gathering (many probable stations)

Random method of matrix computing Peres-Gunnar Martinson

That is, the possibility of an act is equal to the possibility of its components. This departure follows the footprints of the facts of the possibility of empty action (corresponding to event E,).

Randomized numerical line algebra 3. Discrete probability This section adopts an educational program with a simple discrete probability. A more difficult result (especially inequality in the similarity of Bernstein to the probability of the substance and the probability of the matrix) is shown in the later context of this chapter. It is worth noting that Landra's huge share (not continuous probability) is based on a simple and basic principle of separation probability. 3. 1. Random experiment: Lower random experiences are all steps, including a specific set of stochastic results, with infinity. Bone play and coin throwing are common examples. The trajectory of random experience is a large number of stochastic results of random experience. This is often referred to as a random experience (for example, victory and failure). In the doctrine of discrete probability, the location of the sample ω is considered finite. (In this chapter, we do not consider the place of samples or countless places.) The event is an arbitrary part of the Sample Ω location. Definitely, many of the probable events are considered to be a gathering (many probable stations)

If E is an event, (3. 1. 1)

That is, the possibility of an act is equal to the possibility of its components. This departure follows the footprints of the facts of the possibility of the action of the sky (corresponding to the event E,).

Randomization numeric line algebraic lecture

23 22 21 20 19 18

There is an event. In fact, it is possible to justify PR E¯ = 1-PR [E]. In this case, E¯ means the addition to the action of E. Eventually, if E1 is considered an E2 part set, it will be PR [E1] PR [E2]. 3. 3. Restrictions on association. The limit of associations is considered to be the basic result in the discrete probability theory, and has the potential to be applied to restrict the probability of a binding scale without special guess. EI = 1,. During this time, the restrictions on associations are actually claimed as N N EI PR [EI]. PR I = 1

The proof of the unification aspect is very simple, and it may be possible to perform the induction of the two sets removed in the previous section, including the principle of the inclusion of the two sets and exhaust exclusion. 3. 4. Inseparable and autonomous acts. If the two action E1 and E2 intersect, that is, if the E1 ∩ E2 = ∅, it will be no n-consumer or exclusive with each other. This can be summarized for any number of events if all actions are requested to be a pair that does not overlap. Two behavior E1 and E2 are called autonomous, and one of them does not affect another possibility. In fact, these need to meet the conditions of PR [E1∩e2] = PR [E1] -PR [E2]. Again, it is possible to summarize two or more actions so that all actions are autonomous. 3. 5. Relative probability. The relative probability of PR [E1 | E2] for any two events E1 and E2 shows the possibility of E1 on the condition that E2 actually occurs. In effect, PR [E1∩E2] PR [E1 | E2] =. PR [E2]

Petros Dorinius, Michael V. Mahoni

There is no doubt that the possibility of E2 of the denominator must be zero, so it is perfectly defined. According to the wel l-known Bayes' law, in fact, any 2 and e 2-events are actually PR [E1] & amp; gt; gt; 0 and PR [E2] & amp; gt In the; 0, PR [E1 | E2] PR [E2 | E1] =. PR [E1] and the precedent of PR [E1] are actually the position of ω in the part. = It can be drawn as = E2∪ E2, and it is actually in the footprints, as follows.

PR [E1] = PR [E1 | E2] PR [E2] + PR E1 | E2 PR E2. We need to pose with both E1 and E2 probability in the open break (0, 1). You can see that there is. Here, you can return to an independent event. Indeed, for any 2 and E 2-events, they are actually PR [E1] & amp; amp; gt; 0 and PR [E2] & amp; gt; 0, and the appropriate description is equivalent. One: (1) PR [E1 | E2] = PR [E1], (2) PR [E2], (3) PR [E1∩E2] = PR [E1] PR [E2] Remember that the last sentence was the definition of independence in the previous section. 3. 6. Random variables. The random number value is a function that represents the position of the sample Ω to the material amount. Note that it is called a variable, but is actually considered a function. Ω is the place to select random experience. If α∈R is a substance amount (does not need to be positive), the function X-1 (α) = part-part set Ω, and the returned one is considered an event. That is, the X-1 (α) function has a possibility. Scratch by misusing denomi: PR [x = α]

Random method of matrix computing Peres-Gunnar Martinson

PR [X α] = PR X-1 α: α∈ (-UR, α ∈] = PR [ω ∈ ω: x (ω) α]. In relation to the random number value (MFV) and the cumulative divergent function (FFR), this possibility, in fact, the random number size is specific, and the second is the second. It is a possibility of such a thing, and in fact, the random number is observed for any meaning of α∈R (huge probability density function (MFV)).

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Decision 3. 7. 2 (Distribution cumulative function (CDF)). For any value of x and real quantity α, the function F (α) = PR [x α] is called the distribution cumulative function (CDF). From the above definition, it is certain that f (α) = xα f (x). 3. 8. Autonomous random values. Following the view of the independence of events, we can now qualify the notion of independence of random quantities. Indeed, two probability values x and y are considered to be independent if, for all probabilities a and b, pr [x = a and y = b] = pr [x = a] - pr [y = b]. 3. 9. Expectation of a random value If a random measure x is set, the expectation E [x] is oriented as x - pr [x = x]. E [x] = xicle (ω)

In the provided case, x (ω) is a kind of random variable x at the location of the sample ω. That is, the quantity is in the spectrum of the random variable X. As a candidate E [x], it is possible to express it in the definition of the quantity of the area of x. For the final investigation of a place ω as appears in the discrete doctrine of possibility, we receive x (ω) PR [ω]. E [x] = ωicle

Random method of matrix computing Peres-Gunnar Martinson

Petros Dorinias and Michael V. Mahoney

This is definitely following in their footsteps.

In fact, this happens enough to get the dispersed top line. The difference from the waist is that if only random values are considered to be autonomous, dispersion does not have a linear nature. In fact, if the random values of x and y are autonomous, VAR [x + y] = var [x] + var [y]. The above properties are summarized in two or more turbulent numbers, imagining that all participants of the random number are autonomous in pairs. Apart from this, VAR [λx] = λ2 var [x] for any real number λ. 3. 11. Markov inequality. α & amp; amp; gt; 0,

X is a random number that is not an unusual random number.

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T, E [X] for Each Tamp; gt; 0. In order to demonstrate the above inequality, the appropriate function 1 in the case of x T F (x) = 0 is determined by another K =.

Random method of matrix computing Peres-Gunnar Martinson

Note that the prison in this section can use Marcov's inequality to limit the expected value of 4. 1. 2 to limit the expected value of 4. 1. 2, and that the A B-C R-C R-line flobenius Norm is created with immutable possibilities. We (in the sense that Markov's inequality is more dependent on failure than inequality), which is a more serious analysis that focuses the average of the average consent of the queue to support Martingara's discussion. , [9, section 4.

Petros Dorinius, Michael V. Mahoni

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βa ∗ k 2 bk ∗ 2 pk = 1 and pk n, k = 1 a ∗ k 2 bk ∗ 2

During this time, N2 1 (4. 2. 2) E A B-CR2F A * K 2 BK * 2. βC K = 1 is satisfied with the positive constant β 1.

Almost appropriate probability that depends only on A: (4. 2. 3) is satisfied.

In the case of positive constant β 1, 1 A2F B2f. (4. 2. 4) E-A B-CR2F βC B Practical Proper Proper Probability: N K = 1 (4. 2. 5) satisfies (4. 2. 5). 。

When satisfying, 1 (4. 2. 6) e a b-CR2F A2F B2f. ΒC In fact, from the inequality of the Cocary Schwarz, N 2 A2F B2F, K = 1 Be careful.

Therefore, the equation (4. 2. 2) is limited in general than (4. 2. 4) equation (4. 2. 4) or (4. 2. 6). See [9, Section 4. 3, Table 1] for other selection probability and the right mismenargin with a high possibility. 4. 3. Tw o-scale restrictions of two scale restrictions In this section, in both the RandmatrixMulti style methods (see the approximation of the minimum square method of segment 5 and 6, and the approximation of the lo w-ranked matrix) We are interested in the approximation of the highest and most elegant queues, the approximation of the UT U, the highest and most elegant queue, depending on the method of selecting some lines U (and scaling configuration). (A matrix U is a matrix that covers the column space, or a matrix that covers the "important" share of the column space in another matrix that we are interested in.) When u∈Rn x d (nd) is a matrix with regular ort h-line (that is, UT U = ID), it was found that it was possible to focus on personal cases without losing generality. During this time, if you select R∈RC x D

Random method of matrix computing Peres-Gunnar Martinson

Note that the corresponding C (correction) column UT for form a matrix rt from the C (correction) line U built using the Randmatrixmultiply method. (4. 3. 1) Eut U-RT R2F = E I D-RT R2F βc The above is a precedent called U2F = D.

It is sufficient to apply a selection probability PK (k = 1, ..., n) to satisfy (these amounts are common as leverage ratio [17] and are given by equation (4. 3. 2). The probability is in the meaning of the equation (4. 2. 1), that is, in the sense that it is an appropriate probability to proxify the matrix product indicated in the equation (4. 3. 1) with coefficient β. It is thought to be). Using the Markov inequality to the limit of the formula (4. 3. 1), and specifying 10d2 and β 2, (4. 3. 3. 3) can be obtained at least for the probability of at least 9/10.

UT U-RT RF = ID-RT RF.

In fact, it is certain that the bisection remains in the footprint of the given equation. In fact, at least not in 9/10, ut u - rt r2 = id - rt r2 is a device C in the sense of (4. 3. 3). In the remainder of this section, we formulate and prove the axioms that are still possible by ut u - rt r2, setting the sense smaller than that of (4. 3. 3). For a suitable condensation method, see Barshinin's head in this volume [28]. Axiom 4. Let u∈Rn × D (n d) satisfy ut u = id. Construct R with the support of the edge MatrixMultiply method and let the probability of the collection n k = 1 satisfy all k = 1, . . . (4. 3. 6) C β 2 β 2 δ probability is at least 1 - δ but not u - rt r2 = id - rt r2.

Petros Dorinias and Michael V. Mahoney

Lemma 4. 3. 7. X1, X2, ..., XC - D-mortal random vectors X C and E XXT 2 autonomous identically distributed copies of 1. x2 m between which each α & amp; gt; 0,

1 i it x x - e xxt 2 α c c c c

This is implemented with probability at least 2 1 - 2c exp - - but not u - rt r2.

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1 tt t y y. C

Let y 1 m y2 = √ uk ＊ 2 for this vector. PK

Note that in fact, from equation (4. 3. 8), we immediately get E yt y 2 = ID 2 = 1. Theorem 4. 3. 7 (x=yt) gives us (4. 3. 9).

RT R-UT U2 0, LN algorithm (1/Δ)/LN (5) Repeat. In addition, if n is not the following DEUCE, all lines A and line B can be filled in 0 to satisfy the assumption. Note 5. Assuming D N ED, if Max A1 + A2 is used, it becomes D LN N. R = OD (LN D) (LN N) +.

Random method of matrix computing Peres-Gunnar Martinson

Assuming n/ ln n = ω (d2), the above operating time is shortened to D Nd Ln D. O ND LN + FIXED, and the standard O (ND2) operating time of the conventional decisive algorithm is improved. do. It is noteworthy that improvements to the standard O (ND2) time can be obtained even with a weaker assumption of N and D. Note 5. MATRIX ST HD can be considered in one of the following two equivalent methods: as a random count that "unifies" the rating of the lever in input matrix A (accurate of Lemma 5. 4. 1) Refer to the explanation), followed by a single specimen operation, or not only a set of discrete points, but also a random project of Johnson-LindenStraus style that holds the geometric shape of the entire space A (Lemm 5. 4. 5 See accurate explanation). 5. 3. Return to the Landonra algorithm as a spare, laundry strip algorithm is provided by using X-thoroughly designed by random matrix X data. Remember what you can think of (the analysis of our main algorithm for the approximation of the lo w-optional matrix, the analysis of the Land Laurent Coulgorithm is as follows).

Z ~ = min X (A X-b) 2. X route

It is clear that the solution of this problem is calculated using the conventional deterministic algorithm. < SPAN> RT R-UT U2 0, LN algorithm (1/Δ)/LN (5) Repeat. In addition, if n is not the following DEUCE, all lines A and line B can be filled in 0 to satisfy the assumption. Note 5. Assuming D N ED, if Max A1 + A2 is used, it becomes D LN N. R = OD (LN D) (LN N) +.

Randomatic numerical value line algebra

Assuming n/ ln n = ω (d2), the above operating time is shortened to D Nd Ln D. O ND LN + FIXED, and the standard O (ND2) operating time of the conventional decisive algorithm is improved. do. It is noteworthy that improvements to the standard O (ND2) time can be obtained even with a weaker assumption of N and D. Note 5. MATRIX ST HD can be considered in one of the following two equivalent methods: as a random count that "unifies" the rating of the lever in input matrix A (accurate of Lemma 5. 4. 1) Refer to the explanation), followed by a single specimen operation, or not only a set of discrete points, but also a random project of Johnson-LindenStraus style that holds the geometric shape of the entire space A (Lemm 5. 4. 5 See accurate explanation). 5. 3. Return to the Landonra algorithm as a spare, laundry strip algorithm is provided by using X-thoroughly designed by random matrix X data. Remember what you can think of (the analysis of our main algorithm for the approximation of the lo w-optional matrix, the analysis of the Land Laurent Coulgorithm is as follows).

Z ~ = min X (A X-b) 2. X route

It is clear that the solution of this problem is calculated using the conventional deterministic algorithm. RT R-UT U2 0, LN algorithm (1/Δ)/LN (5) Repeat. In addition, if n is not the following DEUCE, all lines A and line B can be filled in 0 to satisfy the assumption. Note 5. Assuming D N ED, if Max A1 + A2 is used, it becomes D LN N. R = OD (LN D) (LN N) +.

Randomatic numerical value line algebra

Assuming n/ ln n = ω (d2), the above operating time is shortened to D Nd Ln D. O ND LN + FIXED, and the standard O (ND2) operating time of the conventional decisive algorithm is improved. do. It is noteworthy that improvements to the standard O (ND2) time can be obtained even with a weaker assumption of N and D. Note 5. MATRIX ST HD can be considered in one of the following two equivalent methods: as a random count that "unifies" the rating of the lever in input matrix A (accurate of Lemma 5. 4. 1) Refer to the explanation), followed by a single specimen operation, or not only a set of discrete points, but also a random project of Johnson-LindenStraus style that holds the geometric shape of the entire space A (Lemm 5. 4. 5 See accurate explanation). 5. 3. Return to the Landonra algorithm as a spare, laundry strip algorithm is provided by using X-thoroughly designed by random matrix X data. Remember what you can think of (the analysis of our main algorithm for the approximation of the lo w-optional matrix, the analysis of the Land Laurent Coulgorithm is as follows).

Z ~ = min X (A X-b) 2. X route

23 22 21 20 19 18

As another method, a standard repetitive method such as a normal residual method with a conjunctive gradient can be used, and the equation (5. 3. You can get the approximation of the optimal solution of 1). This is a strategy implemented with a general approach of Blendenpik/LSRN [3]. Here, a lemma that sets the condition of each queue X that the vector of the minimum solution X-OPT problem satisfies the relative boundary of the minimum tw o-conquer from the formula (5. 3. 1) is settled and proven. Fill.

Petros Dorinius, Michael V. Mahoni

SVD A is A = UA σa v TA. Furthermore, in order to make the notation easier, ⊥t b⊥ = u⊥ a UA B represents the part outside the space A on the right vector B: √ (5. 3. 3) σ2min (Xua) 1/ 2; and (5. 3. 4)

UTA XB⊥ 22 Z2 /2,

It is. It is necessary to pay attention to some points in these conditions. √ First, the conditions (5. 3. 3) only insist on σ2i (XUA) 1/2, but all I = 1,. D, our randomized algorithm is all I = 1, .... This corresponds to the facts of I √ i-uta Xua 2 1-1/ 2. -In another method, you can also use a standard repetitive method such as a normal residual method with a conjunctive slope, and the equation between O (κ (XA) RD LN (1/)). 5. 3. 1) You can get the approximation of the optimal solution. This is a strategy implemented with a general approach of Blendenpik/LSRN [3]. Here, a lemma that sets the condition of each queue X that the vector of the minimum solution X-OPT problem satisfies the relative boundary of the minimum tw o-conquer from the formula (5. 3. 1) is settled and proven. Fill.

Random method of matrix computing Peres-Gunnar Martinson

SVD A is A = UA σa v TA. Furthermore, in order to make the notation easier, ⊥t b⊥ = u⊥ a UA B represents a part outside the space A on the right vector B: √ (5. 3. 3) σ2min (Xua) 1/ 2; and (5. 3. 4)

UTA XB⊥ 22 Z2 /2,

It is. It is necessary to pay attention to some points in these conditions. √ First, the conditions (5. 3. 3) only insist on σ2i (XUA) 1/2, but all I = 1,. D, our randomized algorithm is all I = 1, .... This corresponds to the facts of I √ i-uta Xua 2 1-1/ 2. -In another method, you can also use a standard repetitive method such as a normal residual method with a c o-stammed slope, and between O ((XA) RD LN (1/)). You can get the approximation of the optimal solution of 1). This is a strategy implemented with a general approach of Blendenpik/LSRN [3]. Here, a lemma that sets the condition of each queue X that the vector of the minimum solution X-OPT problem satisfies the relative boundary of the minimum tw o-conquer from the formula (5. 3. 1) is settled and proven. Fill.

23 22 21 20 19 18

SVD A is A = UA σa v TA. Furthermore, in order to make the notation easier, ⊥t b⊥ = u⊥ a UA B represents the part outside the space A on the right vector B: √ (5. 3. 3) σ2min (Xua) 1/ 2; and (5. 3. 4)

UTA XB⊥ 22 Z2 /2,

It is. It is necessary to pay attention to some points in these conditions. √ First, the conditions (5. 3. 3) only insist on σ2i (XUA) 1/2, but all I = 1,. D, our randomized algorithm is all I = 1, .... This corresponds to the facts of I √ i-uta Xua 2 1-1/ 2. -

Therefore, Xua should be considered as a proximity equality. Second, the Lemma is a deterministic statement, since it makes no explicit reference to a particular randomization algorithm and does not assume that X is constructed from a randomization process. The probability of failure will appear later, when we show that our randomization algorithm constructs X that satisfies conditions (5. 3. 3) and (5. 3. 4) with some probability. Third, conditions (5. 3. 3) and (5. 3. 4) determine what has come to be known as the subspace investment, since it is an investment that preserves the geometric shape of the entire subspace of matrix A. This subspace investment can be oblivious (meaning it is constructed without knowledge of the input matrix, as in the random projection algorithm) or non-oblivious (meaning it is constructed based on knowledge of the input matrix, as in the data-dependent unequal sample algorithm). This style of analysis was an important advance in the Randonnéra algorithm, allowing us to obtain much stronger bounds than were possible with previous techniques. For the original paper in which this analysis was used, see [11] (which combines and extends two earlier conference papers). Fourth, given the condition (5. 3

Random method of matrix computing Peres-Gunnar Martinson

Lecture on Randomized Numerical Linear Algebra

The problem of approximation by least squares (5. 3. 1) satisfies (1 +) z and √ 1 xopt - x ˜ Opt 2 (5. 3. 7) Z. σmin (a) Proof. First, rewrite the problem of reducing the recurrence induced by X as Ax~ Opt - B2.

min xa (xopt + y) - x (axopt + b⊥) 22

Min Xua Z - Xb⊥ 22.

From the equation (5. 3. 8), B = axopt + B⊥ is established. In addition, from the equation (5. 3. 9), the column A covers the same part space as the column UA. Here, zopt∈rd that is UA Zopt = a (x˜ op t-xopt). Apply this meaning to Zopt to prove that Zopt is considered to minimize the optimization problem in the following: Xua Zop t-XB⊥ 22 (5. 3. 3. 10)

US Congress Library Catalog Inn publication Data Name Mahoney, Michael W., Editor. | Duchi, John, Editor. | | Publisher name: Mathematics Institute Park City. | Park City. Title: Data Mathematics / Michael V. MAHONI, John K. Duchi, Anna K. Gilbert, Editors. Park City Mathematics Series; Volume 25 | "Institute for Advanced Study "|" Industrial Mathematics Association "| | Includes a book magazine link. ID: lccn 2018024239 | ISBN 9781470435752 (Alk. Paper) Topics: LCSH: Mathematics-Teacher preparation-Conference. | Mathematics research and education conference | Big data conference. | AMS: Linear algebra and multiple algebra; proclaimed theory-Research presentation (book, total theory). MSC ｜ Ret trip geometry and discrete geometry-Research presentation (book / total theory). MSC ｜ Probability theory / probability process theory-Research presentation (book / total theory). MSC | Statistics-Scientific commentary (monograph, general theory). MSC ｜ Numerical Analysis-Scientific Commentary (Monograph, Review Article)

XAX ~ Op t-XAXOP T-XB⊥ 22

Min Xua Z-XB⊥ 22.

The equation (5. 3. 10) follows the footprints of B = axopt + B⊥, and the last equation is led by the equation (5. 3. 9). Therefore, from the normal formula (5. 0. 3), (XUA) T Xua Zopt = (Xua) T XB⊥. Note that if you hold the norm on both sides, and, if you actually have ((XUA) t Xua) = σ2i (XUA) 1/2 for all II σIi, according to the conditions (5. 3. 3), it is the case. From footprints, it actually is as follows (5. 3. 11).

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

In fact, if the conditions (5. 3. 4) are used, Zopt 22Z2 will be used.

Rewrite the residual vector norm to B-A X-Opt 22 to introduce the first claim of the lemma.

B-AXOPT + AXOP T-A X-Opt 22

B-AXOPT 22 + AXOP T-AX ~ OPT 22

Z2+ -UA Zopt 22

Petros Dorinius, Michael W. Mahony

Here, the formula (5. 3. 13) follows the footprint of Pythagoras's axiom. For example, B-Axopt = b⊥ is orthogonal to A, so A (xop t-x˜ opt); equation (5. 3. 14) follows the footprints of Zopt and Z; equation (5. 3. 15) follows the precedent footprint that (5. 3. 12) and UA √ include regular cros s-joints. The first paragraph of the lema reminds me of A (Xop t-X˜ Opt) = UA Zopt to introduce the second description of 1 + 1 +. In fact, if you understand the norm on both sides of this equation, you will get (5. 3. 16).

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

Here, the formula (5. 3. 16) is guided by the fact that σmin (a) is the minimum specific value A and the rank a d is D. In addition, the formula (5. 3. 17) is guided from the orthogonalization of the formula (5. 3. 12) and the U A-column. The second sentence of Lemma removes the square root and is as follows. If you do not assume anything about B, the Lemma 5. 3. 5 formula (5. 3. 7) will hold. 3. 5 gives a weak boundary from the viewpoint of Xopt 2. In contrast, the formula (5. 3. 7) can be enhanced by the additional assumption that a certain share of standard B is located in a partial space covered by column A. Such assumptions are reasonable. This is because most tasks with the minimum square are practically interesting if they are at least a part of the part of B in a column A. 3. 18. Using a legend 5. 3. 5 and assuming UA UTA B2 γB2, √ (5. 3 19) Xop t-X -OPT 2 κ (5. 3. 19) to a fixed γ∈ (0, 1]. a) γ-2-1xopt 2.

B2 2-UA UTA B22

( γ-2-1) UA UTA B22 σ2max (a) ( γ-2-1) Xopt 22. The latest inequality is led by UA UTA B = AXOPT, UA UTA B2 = Axopt 2 A2 Xopt 2 = σmax (A) Xopt 2. When this is combined with the equation of 5. 3. 5 (5. 3. 7), a lemma is obtained. 3. Remma is obtained when combined with the equation of 5 (5. 3. 7).

5. 4. Proof of the theorem 5. 2. 2. In order to prove the theorem 5. 2. 2, we use the following approach: First, the random conversion of Hadamar is almost uniform as a result of export or provisional air conditioning in the input line. Indicates that the conditions (5. 3. 3) and (5. 3. 4) are satisfied by a uniform sampling with a pr e-conditional input. The theorem is Lemma 5. 3. 5.

Petros Dorinius, Michael W. Mahony

B2 2-UA UTA B22

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

5. 4. Proof of theorem 5. 2. 2. In order to prove the theorem 5. 2. 2, we use the following approach: First, the random conversion of Hadamar is almost uniform as a result of export or provisional air conditioning in the input line. Indicates that the conditions (5. 3. 3) and (5. 3. 4) are satisfied by a uniform sampling with a pr e-conditional input. The theorem is Lemma 5. 3. 5.

Randomized numeric lin e-type agency In this case, the formula (5. 3. 16) is guided by the fact that σmin (a) is the minimum specific value A and rank a d is D. In addition, the formula (5. 3. 17) is guided from the orthogonalization of the formula (5. 3. 12) and the U A-column. The second sentence of Lemma removes the square root and is as follows. If you do not assume anything about B, the Lemma 5. 3. 5 formula (5. 3. 7) will hold. 3. 5 gives a weak boundary from the viewpoint of Xopt 2. In contrast, the formula (5. 3. 7) can be enhanced by the additional assumption that a certain share of standard B is located in a partial space covered by column A. Such assumptions are reasonable. This is because most tasks with the minimum square are practically interesting if they are at least a part of the part of B in a column A. 3. 18. Using a legend 5. 3. 5 and assuming UA UTA B2 γB2, √ (5. 3 19) Xop t-X -OPT 2 κ (5. 3. 19) to a fixed γ∈ (0, 1]. a) γ-2-1xopt 2.

B2 2-UA UTA B22

Petros Dorinius, Michael W. Mahony

5. 4. Proof of theorem 5. 2. 2. In order to prove the theorem 5. 2. 2, we use the following approach: First, the random conversion of Hadamar is almost uniform as a result of export or provisional air conditioning in the input line. Indicates that the conditions (5. 3. 3) and (5. 3. 4) are satisfied by a uniform sampling with a pr e-conditional input. The theorem is Lemma 5. 3. 5.

Randomization numeric line algebra

The effect of random reconstruction Hadamard. Left-winged pole partial space "Left information is in the left" wing pole partial space. The rhenma that the proposed method is actually used is UT U = ID, which is actually UT U = ID using the randomization results of the randomization of the fourth matrix. Remma 5. 1. Use the random reconstruction of the Hadamal of the section 5. 1 (5. 1. . 4. 2), all I = 1,. (HDU) I ∗ 22 n The wel l-known inequal type HEID [15, axim 2] is required (obtaining the correct results. See Vershine's head [28] 5. 4. 3. Xi, I = 1, ... N-The last, these, all, all AI Xi Bi. During the autonomous random value, NT 2 N2 T2 X I-E. 2 I = 1 (A I-Bi) I = 1 for any T & AMP; G; 0.

Confirm the lemma 5 on the premise of this lemma. 4. 1. Confirm. (OF Lemma 5. 4. 1) See I and J (HDU) IJ (I = 1,., N, j = 1,., D). During this time, (HDU) IJ =.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

1 | x | = d Hi uj √ uj. Remma 5. 4. 3 can be obtained from 2N3 T2 3 2 = 2 EX P-N T /2. PR (HDU) IJ NT 2 EX P-n 4 = 1 U2J 2 In the last equation, in the precedent, in fact, n = 1 uj = 1, that is, the column U has a single length. there was. If the appropriate percentage of inequality above is δ and T is determined, 2 LN (2/) δ is obtained. PR (HDU) IJ N, δ = 1/(20nd), and use the ultimate associations in all NDs in a high probability of collision (I, j), all I, J, 2 LN (40nd). On the other hand, it is actually obtained that it is not a minimum of 1-1/20 = 0, 95. (Hu) IJ N

Petros Dorinius and Michael V. Mahoni

All I = 1, .. n, actually ends with the confirmation of the lemma.

Conditions (5. 3. 3). Next, prove the following lemma that claims that all the coordinates-values of ST. HDUA are close to 1 and the conditions (5. 3. 3) are realized by the Landrest 2-square algorithm. This theorem 5. 4. 5 is essentially derived from the results of the Randmatrixmumply algorithm (approximation of the procession and its transfer work) of 4. 3. 5 theorem. The theorem 5. 4. 5. Equation (5. 4. 2) is implemented. (5. 4. 6) R 482 D LN (40nd) LN 1002 D LN (40nd), at least 95, 1 - σ2i ST HDUA 1 -√ 2 is all I = 1,. (Lemma 5. 4. 5) All I = 1,. .... 1, 1 - σ2i ST hdua = σi Uta Uta Uta DHT SST HDUA (5. 4. 7) UTA DHT HDU A-UTA DHT SSTUA 2. Above Then, the uta DHT HDUA = ID and the inequality (2. 6. 2) described in verse 2. 6 of the linear algebra were used. Here, we consider the UTA DHT SST HDUA as the approximation of the two queues built on random selections and column scale (HDUA), that is, the approximation of UTA DHT = (HDUA) T and HDUA. Specifically, consider the matrix (HDUA) T. H, D, √ U A-Incorrect, hdua 2 = 1, hdua f = ua f = d.

(HDUA) I * 22 1 β, HDUA 2F

All I = 1, ..., N.

√ Considering β = (2 LN (40nd)) using the ace 4. 3. 5, it becomes 1, = 1-1/2, δ = 1/20. -1, = 1-1/2 and δ = 1/20 are 1 uta DHT HU A-UTA DHT SSTUA 2 1 -√ 2 at least 1-1/20 =. 95. To execute the above restrictions. , R to extract values from equation (5. 4. 6) is required. Note that SINC E HDUA 2F = D 1 is the last.

Petros Dorinius, Michael W. Mahony

The input matrix is satisfied every time. The confirmation of the lemma is completed by the above group and inequality (5. 4. 8). Monkey conditions (5. 3. 4). Next, prove the correct lemma that the condition (5. 3. 4) is established by a laundry strip. Confirmation of this lema 5. 4. 9 is essentially led by the constraints of Randmatrixmultiply in Section 4 (but actually used for the approximation of a matrix and vector). The theorem 5. 4. 9. Equation (5. 4. 2) is actually generated. R 40d LN (40nd) /, at least. 9, T HDUA ST HDB⊥ 22 Z2 / 2. T

⊥ Confirm. (From Lemma 5. 4. 9) Remember that B⊥ = U⊥ A UA B, which is actually Z = B 2. In fact, since UTA DHT HDB⊥ 22 = UTA B⊥ 22 = 0, this follows footprints, actually T HDUA ST HDB⊥22 = UTA DHT SST HDB⊥ -UTA DHT HDB⊥22. In this way, ST HDUA ST HDB⊥ is considered to be an approximate work of (HDUA) T and HDB⊥, which uses an arbitrary column of (HDUA) T and an arbitrary line (element) of HDB⊥. You can. We find that the collection probability is uniform, and it does not depend on the rows of HDUA, which are generally recognized (HDUA) or HB⊥ line. Next, a restriction formula (4. 2. 4) is applied so that the guess of the formula (4. 2. 3) is executed. Indeed, the HDUA line (of course that fills the row (HDUA) T) meets the following conditions, so that the execution of the formula (5. 4. 2) is determined.

(HDUA) I * 22 1, β HDUA 2F

All I = 1, ..., N,

β = (2LN (40nd)) -1. In this way, from the footprint formula (4. 2. 4), it actually becomes T 1 DZ2 T ⊥ 2 E S HDUA S HDB 2 HDB⊥ 22 =. ββR We used HDUA 2F = D. By the way, from the inequality of the footprint Marcov, at least 9, t 10dz2. ST HDUA ST HDB⊥ 22 βR is used to R 20D/(β), and the value indicated by higher β is used, so Confirmation is completed. The confirmation of the axiom 5 is completed. The axiom is in the footprints of Remma 5. 4. 5 and Remma 5. 4. 9. This is because the conditions of 5 are sufficiently established. Here, even for e (5. 4. 2) de 5. 2. 2., the check of the starter's axiom is completed. Note that the formula (5. 4. 2) is executed. Definitely PR E (5. 4. 2). 95.

Petros Dorinius and Michael V. Mahoni

E5. 4. 5. 4. 9 ｜ (5. 4. 2), Rem 5. 5 and 5. 9 9 under E (5. 4. 2) It means being executed under .. 2). And E5. 4. 5. 4. 9 | (5. 4. 2) = 1-E5. 4. 5. 4. 9 | (5. 4. 2) = 1-PR Lemma 5. 4. 5 Is not implemented 5. 4. 9 is not implemented | E (5. 4. 2) 1-PR Lemma 5. 4. 4. 5 does not prefer 1. 1 =. 85. In the above example, E means the addition of Elephant E. In the first inequality, the ultimate of associations is used, and the second inequality forms a Lemm 5. 5 and 5. 4. 9 formula (5. 4. 2). Here, suppose that gives an event when the equation (5. 4. 2) is executed. When both 5 and 5. 4. 9 are satisfied, PR [E] is restricted as follows: PR [E] = PR E | E (5. 4 (5. 4) . 2) -PR E (5. 4. 2) + PR E | E (E (5. 4. 2)) -PR E (5. 4. 2) = PR E5. | (5. 4. 2) | 95. 8. In the first inequality, we used the fact that all probability was positive. The above conclusion is to limit the success probability immediately by the theorem 5. 2. 2. The theorem 5. 5 and 5. 4. 9 are combined with the structural result of [5. 3. 5]. By combining 3. 5 and configuring R as an equation (5. 2. 3), the accuracy guarantee of the theorem 5. 2. 2 is completed. 5. Laundry Street Square Algorithm's operating hours Next, define the operating time of the laundry strip algorithm.

Rem 5. 5 and 5. 4. It means that 9 is executed under E (5. 4.. 2). And E5. 4. 5. 4. 9 | (5. 4. 2) = 1-E5. 4. 5. 4. 9 | (5. 4. 2) = 1-PR Lemma 5. 4. 5 Is not implemented 5. 4. 9 is not implemented | E (5. 4. 2) 1-PR Lemma 5. 4. 4. 5 does not prefer 1. 1 =. 85. In the above example, E means the addition of Elephant E. In the first inequality, the ultimate of associations is used, and the second inequality forms a Lemm 5. 5 and 5. 4. 9 formula (5. 4. 2). Here, suppose that gives an event when the equation (5. 4. 2) is executed. When both 5 and 5. 4. 9 are satisfied, PR [E] is restricted as follows: PR [E] = PR E | E (5. 4 (5. 4) . 2) -PR E (5. 4. 2) + PR E | E (E (5. 4. 2)) -PR E (5. 4. 2) = PR E5. | (5. 4. 2) | 95. 8. In the first inequality, we used the fact that all probability was positive. The above conclusion is to limit the success probability immediately by the theorem 5. 2. 2. The theorem 5. 5 and 5. 4. 9 are combined with the structural result of [5. 3. 5]. By combining 3. 5 and configuring R as an equation (5. 2. 3), the accuracy guarantee of the theorem 5. 2. 2 is completed. 5. Laundry Street Square Algorithm's operating hours Next, define the operating time of the laundry strip algorithm.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

Randomization numeric line algebra

Let T(n) be the number of operations that are important to perform a given operation on a one-dimensional vector. Meanwhile, T(n) = 2T(n/2) + n, i. e., T(n) = O(n log n). Now, by inserting the subsampling matrix S, we get H x1 Hn/2 n/2 = S1 Hn/2 (x1 + x2 ) + S2 Hn/2 (x1 - x2 ). S1 S 2 Hn/ 2-Hn/2 x2 nnz(-) denotes the number of nonzero entries in the argument. Meanwhile, T(n, nnz(S)) = T(n/2, nnz(S1)) + T(n/2, nnz(S2)) + n. Using the usual methods of analyzing recursive algorithms, we can justify that r = nnz(S) = nnz(S1) + nnz(S2), i. e., T(n, r) is 2n log2 (r + 1). To do this, let r1 = nnz(S1) and r2 = nnz(S2). Meanwhile, T(n, r) = T(n/2, r1) + T(n/2, r2) + n n 2 log2(r1 + 1) + 2 log2(r2 + 1) + n log2 2 = n log2(2(r1 + 1)(r2 + 1)) n log2(r + 1)2 = 2n log2(r + 1). The last inequality follows in the footsteps of the simple algebra that introduced r = r1 + r2. Therefore, when using ST HD on d + 1 vectors, no more than 2n(d + 1) log2(r + 1) operations are required. After this preprocessing, the RandLeastSquares method needs to find the pseudoinverse of an r × d matrix, or solve a least-squares problem with r constraints and d variables. This operation can be performed in O(rd2) time, say r d. Thus, the whole method is completed in time n(d + 1) + 2n(d + 1) log2 (r + 1) + Ord2. 5. 6. Co

6. RandNLA Method for Low-Rank Matrix Approximation In this section, we assume a simple randomized matrix method for approximating low-rank matrices. Methods for computing low-rank approximations of matrices have traditionally been central to scientific computing.

Petros Dorinias, Michael W. Mahoney

Petros Dorinius, Michael W. Mahony

K AF (1 +) A-UK UT AF = (1 +) A-AK F. ~ Ku a-U K < Span> See [23] for the numerical methods), and recently the field of machine learning and data analysis. Randnla has become a pioneer in alternative layouts using random collection and random projections to build these lo w-ranking approaches that have been guaranteed accuracy. See [7] for Wound Work for this topic. This section assumes an unprecedented method for approximation of the upper K-left special vector of the matrix A (RM x N). Almost all methods of Randnla for approximation of low ranks are returned to this basic option. Randnla for regression task s-Unlike the previous section on the method, M and N have little special assumptions; this may be a square matrix. 6. 1. Pr e-laws and major axiomes We are very simple, and the random reconstruction of Hadamards in verse 5. 1 is used again. First, correct the input matrix a∈rm x n on (HD) t, and the new matrix ADH∈rm x n. 1 is as follows.

K AF (1 +) A-UK UT AF = (1 +) A-AK F. -KU A-U K See [23] for the method), and recently it is the field of machine learning and data analysis. Randnla has become a pioneer in alternative layouts using random collection and random projections to build these lo w-ranking approaches that have been guaranteed accuracy. See [7] for Wound Work for this topic. This section assumes an unprecedented method for approximation of the upper K-left special vector of the matrix A (RM x N). Almost all methods of Randnla for approximation of low ranks are returned to this basic option. Randnla for regression task s-Unlike the previous section on the method, M and N have little special assumptions; this may be a square matrix. 6. 1. Pr e-laws and major axiomes We are very simple, and the random reconstruction of Hadamards in verse 5. 1 is used again. First, correct the input matrix a∈rm x n on (HD) t, and the new matrix ADH∈rm x n. 1 is as follows.

K AF (1 +) A-UK UT AF = (1 +) a-ku a-u k

(UK∈rm x K has a K-left wing-algebra vector A). The calculation time of the Randlowrank method is O (MNC). 6-9 Determine the dimension of the matrix in the steps of the Randlowrank method. The matrix C∈RM x C can be considered "sketch" in the input matrix A. C can be found to be o (k ln k) (the coefficient of LN LN ignores the constant 1 and instead).

You will have the opportunity to multiply with HD. Readers should be able to easily break from these operations.

Reading related to random numerical ward algebra

The UC matrix has dimensions m × ρc, and the matrix w has dimensions ρc × n. Finally, the matrix UW, k has dimensions ρc × k (due to the rank W assumption). Axiom 6. 1. 1 declares that the Randlowrank method returns a set of K orthonormal vectors. First data: a∈Rm × n, rank parameter K minus error ∈ (0, 1/2). (1)Let C be the meaning of equation (6. 1. 2). (2)Let S be the empty matrix. (3) t = 1 , . . C (Check for I. I. D. permutation)We are in a single quantity randomly selected measured. If I is chosen, n/c is ei of s, and ei∈Rn is the sum of the vector dotz of the i-th canonical vector. (4) Let h∈Rn×n - the normalized matrix of Hadamard rearrangement. (5) d∈Rn × n-diagonal matrix C +1, with 1/2 probability DII = -1, with 1/2 probability (6) Compute C = ADHS∈RM × C (7) Compute UC, the basis for the position of column C. (8) Determine W = UTC A and determine the k-upper-left-row vectors UW, K (assuming that the rank is not actually less than k).

Petros Dorinius at Michael V. Mahoney

Remarks 6. Radlowrank l n-METHOD (1/ Δ)/ ln 5 One and ~ Ku ~ TK AF minimizes the monitoring of queue U (0, 1), and refuses to reserve a queue U. Possibility can be reduced by 1-Δ or less. Remarks 6. 1. 7. Similar to the collection processing in the RandleasTSQUARES method, the operation indicated by the DHS in the Randlowrank method may be considered a project in Johnso n-style LindenStraus. (Especially unofficially, the Randlowrank method works at the correct basis. In the case of the Randleastsquares method, there is a chance to have several columns in advance, and more often converts input data to a new base. Note 6. Increase of (K/ 2) LN N). (With a different C) can also justify the many other random implication algorithms < Span> RADLOWRANK L N-METHOD (1/ Δ)/ Ku ~ TK AF. For each Δ∈ (0, 1), the monitoring of the conservation of the queue U can be minimized, and the possibility of rejection can be reduced to 1-Δ. 7. Similarly, the operations indicated by the DHS in the Randlowrank method may be considered Projection in LindensStraus in the Johnson style (especially in the case of the Randleasquares method. There is a chance to have a column, and in more cases, the highest leverage to convert input data to a new base can be selected 6. Size C can be selected. Invisible to LN N) can be justified as a function KN K and LN N. 6. Radlowrank l n-METHOD (1/ Δ)/ ln 5 One and k u-tk af are repeated, minimizing the preservation of queue U (0, 1), minimizing the preservation of queue U and being rejected. Remarks can be reduced to 1-Δ. 1. 7. In particular, the Randlowrank method works at the correct basis. In the case of the RandleastSquares method, there is a chance to have several columns in advance, and in more cases, the highest leverage to convert input data to a new base can be selected. 6. Size C is equal to o ((k/ 2) ln (k/) ln n). In the case of invariant, it increases as function K ln k and LN n. Such restrictions can be justified on many other random implies algorithms (having different C).

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

Lower the members of LN LN N. Therefore, if you quote Stan Aizenstat, LN LN N is essentially a constant. https: // rjlipton. Wordpress. com/2011/01/19/we-believe-a-Lot-Can-prove-Little/.

Randomization numeric line algebraic lecture

Many lo w-order Landra approximation algorithms and their analysis. In fact, A K-UC UTC AK 2F 2 ( A-AK) DHS ((V Tk DHS) †- (v Tk DHS) T) 2F + 2 ( A-AK) DHS (V Tk DHS) T 2F. Part 3 (Section 6. 4) uses the results of Section 4 to restrict two members on the right side of the above inequality. 6. 2. The alternative expression of errors Randlowrank algorithm is approximated with the upper K's left special vector a, that is, the queue UK RM x K, ON ~ Ku ~ TK OFF, as it appear, ~ k ∈ RM x k. 。 U-procession also prove that the interesting nature of U 3 is the best prospect of A (in accordance with Flobenius Norm) in the line C space. The nature of this optimality is guaranteed by a step like RITZ-RITZ implemented in the 7-9 algorithm Randlowrank ~ K-lemma 6 steps. 2. 1. UC ~ is the basis of the range of column C, and U ~ is the output of the Randlowrank algorithm. And ~ tk a = a-uc ut a. ~ (6. 2. 2)

Furthermore, UC (UTC A) K is the best rank approximation of 1 according to the flovenius standard, and in the row of queue C, that is, (6. 2. 3).

A-UC (UTC A) K 2F =

MIN A-UC Y2f. Rank (Y) K

Here, UW and K are the lines of the upper left K. U-peculiar vector w = UTC A. There are, UW and K include the same range as WK, which is the best rank of W, that is, uw, k utw, k = wk w † k. Therefore, ~ tk a-uw, ku a ~ ku a-u w, ku a-u w, k C c = a-uc wk w † kw = a-uc w k. Guided by being a bowl. To prove the optimality of lemma (lemma.) K 2f = a-uc utc a + utc a-uc (utc A) k 2f = ( i-uc utc) a + uc (UTC A- (UTC A) K) 2F = ( i-UC UTC) A2F + UC (UTC A- (UTC A) K) 2F = ( i-UC UTC) A2F + UTC A- (UTC A) K 2F It is enough to be aware of that. 3 This is

Other Unit Norm, for example, 2-minute is not fair.

Petros Dorinius, Michael W. Mahony

The last equation is derived from the pythagoras matrix (Lemma 2. 5. 2), and the last equation is led by the orthogonalization of the U C-column. This is because (UTC A) K is the best approximation of UTC A Rank K, and therefore other queues of Y, which are not bigger than K, increases the error of Flobenius Norm. Lema 6. 2. 1 indicates that the equation (6. 1. 3) of the theorem 6. 1. 1 can prove using the boundary indicated by A-UC UTC A F. Next, move from the best rank similar K in the projection matrix (UTC A) to the best rank K near the source matrix. First, consider the following (recall the name introduced in section 2. 6 (6. 2. 4)).

Here, ak = uk = uk σk v tk, ak, ⊥ = uk, ⊥ σk, ⊥ v tk, ⊥.

Theorem 6. 2. 5. The regular orthogonal basis for the row of columns from the UC to the K is the output of the Randlowrank algorithm. Next, C, U T

K A2 A K-UC UT AK 2 + AK, ⊥ 2. K a-u c fing certificate. The optimal properties of the equation 6. 2. 1 (6. 2. 3), the optimal properties of the equation of 2. 1. (6. 2. 3), and the UTC AK are less K or less ~ tk a2 = a-uc ut. a 2 ~ ku a-u c fk k kk fk fk fk fk

A-UC UTC AK 2F

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

6. 3. Structural inequality This is a format and prove a structural inequality formula that helps constrain the A K-UC UTC AK 2F (Lemma 6. 2. 5's Miss Limit 1). For the given matrix a∈rm x n, almost all Randnla methods are "sketches" in A∈Rn x C (c is smaller than n) to A to the "sketch" of A. Remember that you want to be configured. The classification of the team is to study how much AZ covers the main part of the main part A, and one of the accuracy indicators is a missing selection Norm of AK- (AZ) (AZ) † AK. It is thought to be there. The following structural results emphasize the probability of constrainting the flovenius norm of AK AK- (AZ) (AZ) † AK. Theorem 6. 3. 3. 1. 1. A∈rm x n, z∈Rn x C (ck) is an arbitrary matrix, which means that V Tk Z∈RK x C contains rank K. During this time (6. 3. 2)

Ak - (az) (az) † AK 2F ( a-ak) z (v tk z) † 2f.

Randomized numeric line algebra lecture < SPAN> 6. 3. Structural inequality type Here, a structural inequality formula that is useful to restrict A K-UC UTC AK 2F (Lemma 6. 2. 5 mislimit). , Prove. For the given matrix a∈rm x n, almost all Randnla methods are "sketches" in A∈Rn x C (c is smaller than n) to A to the "sketch" of A. Remember that you want to be configured. The classification of the team is to study how much AZ covers the main part of the main part A, and one of the accuracy indicators is a missing selection Norm of AK- (AZ) (AZ) † AK. It is thought to be there. The following structural results emphasize the probability of constrainting the flovenius norm of AK AK- (AZ) (AZ) † AK. Theorem 6. 3. 3. 1. 1. A∈rm x n, z∈Rn x C (ck) is an arbitrary matrix, which means that V Tk Z∈RK x C contains rank K. During this time (6. 3. 2)

Petros Dorinius, Michael W. Mahony

Randomized numerical lin e-type algebra 6. 3. Structural inequality type This is a formula and proven a structural inal formula that is useful to restrict A K-UC UTC AK 2F (Lemma 6. 2. 5's Miss Limit 1). Masu. For the given matrix a∈rm x n, almost all Randnla methods are "sketches" in A∈Rn x C (c is smaller than n) to A to the "sketch" of A. Remember that you want to be configured. The classification of the team is to study how much AZ covers the main part of the main part A, and one of the accuracy indicators is a missing selection Norm of AK- (AZ) (AZ) † AK. It is thought to be there. The following structural results emphasize the probability of constrainting the flovenius norm of AK AK- (AZ) (AZ) † AK. Theorem 6. 3. 3. 1. 1. A∈rm x n, z∈Rn x C (ck) is an arbitrary matrix, which means that V Tk Z∈RK x C contains rank K. During this time (6. 3. 2)

Ak - (az) (az) † AK 2F ( a-ak) z (v tk z) † 2f.

Randomization numeric line algebraic lecture

Note 3. 3. 3. The theorem 6. 3. 1 is fair, whether it is determined or random for any matrix Z. In the context of the Randnla, the typical ZER structure is a random sample or random disguise like a DHS matrix used in the Randlowrank algorithm. 6. 4. Lemma is actually fair for any unusual Norma, including the tw o-t o-norma in the line and a nuclear norm. Remarks 6. 3. 5. See [18] for detailed discussions and history of such structural inequality. From 3. 3. 1, a proof strategy to restrict the error of the Randnla algorithm that is approximately the low rank matrix is determined by the ZS sketch that will make V TK Z become full rank, and at the same time † T. Limit the corresponding Norm V K Z Z and ( A-AK) Z. (Lemma 6. 3. 1) First, (AZ) † AK = ARGMINX MOREC × N AK- (AZ) X2F. The above formula is guided by considering the above optimization problem as a regression with a minimum tw o-square method with a different right side. Interestingly, this nature is fair to any unusual Norm, but the proof is very complicated. See 2]. As a result, instead of restricting AK - (AZ) (AZ) † AK 2F, (AZ)+AK can be replaced by other matrix C × n. especially

T = a k-u k-uk σk (v tk z) (v tk z) † σ-1 k UK ak

= A k-UK UTK AK = 0.

In the formula (6. 3. 6), the fact that both the equation (2. 9. 1) and the matrix V Tk Z and UK σk have a rank K. In addition, from the last fact (V Tk Z) (V Tk Z) † = i, the formula flows to (6. 3. 7). Finally, the fact that UK UTK AK = ak is ak completes the conclusions and proof of the lemma.

Petros Dorinias and Michael V. Mahoni < SPAN> Note 6 3. 3. The theorem 6. 3. 1, whether it is a random or random for any matrix Z It is. In the context of the Randnla, the typical ZER structure is a random sample or random disguise like a DHS matrix used in the Randlowrank algorithm. 6. 4. Lemma is actually fair for any unusual Norma, including the tw o-t o-norma in the matrix, norma. [18] Remarks 6. 3. 5. See [18] for detailed discussions and history of such structural inequality. From Remma 6. 3. 1, a proof strategy to limit the error of the Randnla algorithm that is approximately the low rank matrix is determined to be a ZS sketch that will make V TK Z become full rank, and at the same time † T Limit the corresponding Norm V K Z Z and ( A-AK) Z. (Lemma 6. 3. 1) First, (AZ) † AK = ARGMINX MOREC × N AK- (AZ) X2F. The above formula is guided by considering the above optimization problem as a regression with a minimum tw o-square method with a different right side. Interestingly, this nature is fair to any unusual Norm, but the proof is very complicated. See 2]. As a result, instead of restricting AK - (AZ) (AZ) † AK 2F, (AZ)+AK can be replaced by other matrix C × n. especially

Petros Dorinius, Michael W. Mahony

= A k-UK UTK AK = 0.

In the formula (6. 3. 6), the fact that both the equation (2. 9. 1) and the matrix V Tk Z and UK σk have a rank K. In addition, from the last fact (V Tk Z) (V Tk Z) † = i, the formula flows to (6. 3. 7). Finally, the fact that UK UTK AK = ak is ak completes the conclusions and proof of the lemma.

Petros Dorinius and Michael V. Mahoni 6 3. 3. The aor other 6. 3. 1 is fair, whether it is a random or random for any matrix Z. In the context of the Randnla, the typical ZER structure is a random sample or random disguise like a DHS matrix used in the Randlowrank algorithm. 6. 4. Lemma is actually fair for any unusual Norma, including the tw o-t o-norma in the line and a nuclear norm. Remarks 6. 3. 5. See [18] for detailed discussions and history of such structural inequality. From 3. 3. 1, a proof strategy to restrict the error of the Randnla algorithm that is approximately the low rank matrix is determined by the ZS sketch that will make V TK Z become full rank, and at the same time † T. Limit the corresponding Norm V K Z Z and ( A-AK) Z. (Lemma 6. 3. 1) First, (AZ) † AK = ARGMINX MOREC × N AK- (AZ) X2F. The above formula is guided by considering the above optimization problem as a regression with a minimum tw o-square method with a different right side. Interestingly, this nature is fair to any unusual Norm, but the proof is very complicated. See 2]. As a result, instead of restricting AK - (AZ) (AZ) † AK 2F, (AZ)+AK can be replaced by other matrix C × n. especially

T = a k-u k-uk σk (v tk z) (v tk z) † σ-1 k UK ak

= A k-UK UTK AK = 0.

In the formula (6. 3. 6), the fact that both formula (2. 9. 1) and matrix V TK Z and UK σk have rank K. In addition, from the last fact (V Tk Z) (V Tk Z) † = i, the formula flows to (6. 3. 7). Finally, the fact that UK UTK AK = ak is ak completes the conclusions and proof of the lemma.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

6. 4. The end of the positive 6. 1. 1. For this, complete the strategy shown at the end of Section 6. 1 to complete the conditional guarantee of the mistakes of 6. 1. 1. First of all, the theorem is actually linked to the following from the theorem 6. 2. 5 (6. 4. 1).

K A2 A K-UC UT AK 2 + a --AK 2.

In order to combine the first member of the right part of the above inequality, we build a queue D, H, S as described in the Randlowank method, as described in the Randlowank method for lemma 6. 3. do. If v tφ and k s contain rank k, the evaluation of Lemma 6. 3. 1. In fact (6. 4. 2)

Φk - (φS) (φs) φk 2F ( φ-φk) s (v tφ, k s) † 2F.

Here, the matrix of the upper K-right vector φ is used using v φ, k∈RN x k. DH is an orthogonal matrix, and as a result, the meaning of the unique vector of the matrix A and the φ = ADH is similar to the unique vector. The right vector of the matrix φ is a primitive right-wing-peculiar vector rotated on the DH, only V TP = V T DH. ) In this way, φk = ak DH, Φ-φk = ( a-ak) DH, v φ, k = v k DH. If you apply all of the above, you can rewrite the formula (6. 4. 2) as follows: (6. 4. 3)

K A2 A K-UC UT AK 2 + a --AK 2.

According to the above conclusion, D H-members were removed from the universal recognized measurement value of Flobenius, using units of unit. In fact, I want you to recall that it is ak, ⊥ = a-ak. Here, operate the right part of the above inequality in the future method 4: AK, ⊥ DHS (v Tk DHS) † 2F = ak, ⊥ DHS ((v Tk DHS) †- (v Tk DHS) T + ( V Tk DHS) T) T) 2F (6. 4. 4) (6. 4. 5)

2AK, ⊥ DHS ((v Tk DHS) † - (v Tk DHS) T) 2F + 2AK, ⊥ DHS (v Tk DHS) T 2F.

Petros Dorinius, Michael W. Mahony

We use the simple version of the right of the triangular inequality for freezing, which is universally recognized by Flobenius: X + Y for the same dimension of any tw o-line x and y. 2F 2x2F + 2Y2F.

Randomization numeric line algebra

Hadamard effect of random rearrangement. Here we formulate a lemma that defines how HD (premultiply V K, or DH then multiply V Tk), approximately the right information of the singular subspace of matrix A, can be used to allow us to use our results from matrix multiplication from section 4 for the restrictive (6. 4. 4) and (6. 4. 5). This is exactly the same as the consideration in section 5. 4. Lemma 6. 4. 6. Let V K - an N×K matrix with orthogonal columns, and the HD-product is randomized by the Hadamard N×N rearrangement of section 5. 1. During this time, at least . 95, 2k ln (40nk), for all i = 1, . . . N. The verification of the lemma above is the same as the verification of lemma 5. 4. 1. 6. 4. 1. The restriction of equation (6. 4. 4). To restrict the members of equation (6. 4. 4), we first use the universally accepted strong subordination of the Frobenius advantage (see Section 2. 5) to obtain AK, ⊥ DHS ((V TK DHS) † - (v tk dhs) t) t) 2f (6. 4. 8). )

ak, ⊥ dhs2f (v tk dhs) † - (v tk dhs) t 22.

The first lemma constrains the terms (a - ak) dhs2f = ak, ⊥ dhs2f. In fact, we confirm this result for each matrix X and for the choice of matrix S in the Randlowrank method. Theorem 6. 4. 9. Construct the choice matrix S∈Rn×C in the same way as in the Randlowrank method. Meanwhile, for any matrix x∈Rm×n, e xs2f = x2f, and by Markov's inequality (see Section 3. 11), with a probability of at least 0. 95, xs2f 20x2f. Remarks 6. 10. The above lemma is fair, including if the selection of the canonical EI canonical vector is not performed randomly, but by considering each set of possibilities summarized by units, provided that the canonical vector selected on the T-M test (e. g., the canonical vector eit) is scaled by 1/cpit. In this way, including the unequal sample XS, the commonly recognized evaluation of the Frobenius measure of the matrix X is considered. (from Lemma 6. 4. 9) We calculate the expectation of XS2F from the initial basis in the following way: c c n n 1 n 1 - x ∗ j 2f = x ∗ j 2f = x2f. e xs2f = n c c t = 1 j = 1

Now the lemma follows in the footsteps of Markov's inequality.

Petros Dorinias, Michael V. Mahoney

Here, assuming that the equation (6. 4. 7) is implemented, the following lemma can be proved. Assuming that Renma 6. 4. 11. equation (6. 4. 7) is executed. When C is satisfied with √ 192K LN (40NK) 192 20k LN (40NK) (6. 4. 12) C, Ln 2 2, at least probability. 95, † t Tk DH S-v Tk DHS 22 2 2 Suppose it represents a specific value of the matrix V TK DHS. We can repeat the proof of Lemma 5. 4. 5 under the condition of execution of the formula (6. 4. 7). If 4. 5 is repeated and C meets the equation (6. 4. 12), at least 95 probability (6. 4. 13) 1-σ2i is executed for all I (in fact, we You can repeat the proof of Lemma 5. 4. 5). 4. The proof of 5 can be repeated again with V K instead of UA, and K instead of D. ) Note that the matrix † T and V Tk DHS V TK DHS have the same left and righ t-right vectors. In this lemma, T-σi represents the T-K DHS specific value in T σi. Therefore, 2 † t 2- 2-σ v Tk DH S-v tk DHS 22 = max σ-1 - σ = max) σ (1 I i.)

Combined with formula (6. 4. 13) and 1/2, † t-1 2 2 v Tk DH S-v TK DHS 22 = max ( 1-σ2i) σ-2 I (1-) 2.

Petros Dorinius, Michael W. Mahony

Randomization numeric line algebra

Establish our boundaries in section 4. Our reasoning is completely consistent with the proof of [5]. 4. 9. Prove the following lema. Assume that Remma 6. 4. 15. Formation (6. 4. 7) is satisfied. If c40k LN (40NK)/, if you are a . 95 or higher probability, 95, AK, ⊥ DHS (V Tk DHS) T 2F AK, ⊥ 2F. TK DHS) T 2f = AK, ⊥ DHSST HT DV K-AK, ⊥ DHHT DV K 2F, DHHT D = in, AK, ⊥ v K = 0. Therefore, AK, ⊥ DHSST HT DV K can be considered an approximate between two matrix, ak, ⊥ DH and HT DV K. After confirming that the assumption of the equation (4. 2. 5) is satisfied, the restriction of the equation (4. 2. 6) is applied. Since the equation is set as the condition (6. 4. 7), the rows of HT DV K = HDV K are (HDV K) I ∗ 22 1 β for all I = 1, ... 。 β = (2 LN (40NK)) -1. Therefore, from the formula (4. 2. 6), 1 A DH2F HDV K 2F E AK, ⊥ DHSST HT DV K-AK, ⊥ DHHT DV K 2F βC K, ⊥ Ka 2. = Βc K, ⊥ F (6 4. 16)

In the above, HDV K 2F = K was used. Here, the following probability comes from Marcov's inequality. 95, 20k AK, ⊥ 2f. AK, ⊥ DHSST HT DV K-AK, ⊥ DHHHT DV K 2F βC R is 20k/(β), and using the β value described above is completed. Complete the proof of theorem 6. 1. 1. Now that we are ready to complete the proof, we will return to the use of ak, ⊥ = a-ak. First, formulate the following lemma. Assume that you will meet the Rhemma 6. 4. 17. Formation (6. 4. 7). If K Ln Nk Ln N (6. 4. 18) C0 2 LN, 2, with a probability of . 85 or higher, T, ⊥. 85, T

˜ K AF (1 +) A-AK F. ˜ Ku A-U certificate. Renma 6. 2. 5 Type (6. 4. 4) and (6. 4. 5), and from the aorves 6. 14 and 6. 415

K A2 (1 + + + 80 2) a-ak 2 (1 + 41) a-ak 2.

Petros Dorinius and Michael W. Mahony

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

A-uc must be equal to the maximum value used in Rem 6. 4. 14 and 6. 4. 15, this is the value of the formula (6. 4. 12). Remma can be obtained by modifying to 21 and adjusting the corresponding equation C constant. (We do not make any special efforts to calculate or optimize C in C.) The probability of rejection is led by the expression (6. 4. 7) between 6. 14 and 6. 415 (6. 4. 7). To complete the proof of theorem 6. 1. 1, the conditional probability may be deleted from equation (6. 4. 17). 4. 17. To do this, take the same strategy as section 5. 4 and conclude that the probability of success of the general approach is at least 0, 85-0, 95 0, 8. 6. 5. Opening hours 6. 1. 4 algorithm Randlowrank calculates the accumulation C = AHDS using the idea of section 5. 5, and takes time for 2n (m + 1) log2 (c + 1). Step 7 is o (mc2); Step 8 is o (mnc + nc2); step 9 is O (MCK). Generally, as a perspective, the operation time is dominated by the O (MNC) and C C C C in step 8, as in the formula (6. 4. 18). 6. 6. The description in the link this section follows the conclusion of [8]. Also, please refer to [21, 29] for readers who are interested. Thank you. The author is grateful for the permission of the ILZE IPSEN slide. < SPAN> a-uc must be equal to the maximum value used in Rem 6. 4. 14 and 6. 4. 15, this is the value of the formula (6. 4. 12). Remma can be obtained by modifying to 21 and adjusting the corresponding equation C constant. (We do not make any special efforts to calculate or optimize C in C.) The probability of rejection is led by the expression (6. 4. 7) between 6. 14 and 6. 415 (6. 4. 7). To complete the proof of theorem 6. 1. 1, the conditional probability may be deleted from equation (6. 4. 17). 4. 17. To do this, take the same strategy as section 5. 4 and conclude that the probability of success of the general approach is at least 0, 85-0, 95 0, 8. 6. 5. Opening hours 6. 1. 4 algorithm Randlowrank calculates the accumulation C = AHDS using the idea of section 5. 5, and takes time for 2n (m + 1) log2 (c + 1). Step 7 is o (mc2); Step 8 is o (mnc + nc2); step 9 is O (MCK). Generally, as a perspective, the operation time is dominated by the O (MNC) and C C C C in step 8, as in the formula (6. 4. 18). 6. 6. The description in the link this section follows the conclusion of [8]. Also, please refer to [21, 29] for readers who are interested. Thank you. The author is grateful for the permission of the ILZE IPSEN slide. A-uc must be equal to the maximum value used in Rem 6. 4. 14 and 6. 4. 15, this is the value of the formula (6. 4. 12). Remma can be obtained by modifying to 21 and adjusting the corresponding equation C constant. (We do not make any special efforts to calculate or optimize C in C.) The probability of rejection is led by the expression (6. 4. 7) between 6. 14 and 6. 415 (6. 4. 7). To complete the proof of theorem 6. 1. 1, the conditional probability may be deleted from equation (6. 4. 17). 4. 17. To do this, take the same strategy as section 5. 4 and conclude that the probability of success of the general approach is at least 0, 85-0, 95 0, 8. 6. 5. Opening hours 6. 1. 4 algorithm Randlowrank calculates the accumulation C = AHDS using the idea of section 5. 5, and takes time for 2n (m + 1) log2 (c + 1). Step 7 is o (mc2); Step 8 is o (mnc + nc2); step 9 is O (MCK). Generally, as a perspective, the operation time is dominated by the O (MNC) and C C C C in step 8, as in the formula (6. 4. 18). 6. 6. The description in the link this section follows the conclusion of [8]. Also, please refer to [21, 29] for readers who are interested. Thank you. The author is grateful for the permission of the ILZE IPSEN slide.

(1) N. Iron and B. Shazel, high-speed Johnson-Linden Strauss Conversion Recently, Siam J. Comput. 39 (2009), No. 1, 302-322, Doi 10. MR2506527 ← 25 [2] N AILON and E. Liberty, Fast Dimension Redemente r-Series on Dual Bc h-Codes, Discrete Comput. 42 (2009), DOI 10. 6458 ← 35 [3] H. AVron , MAYMOUNKOV AND S. TOLENPIK: SuperCharging Leapack's Solver, Siam J. Sci. 32 (2010), DOI 10. 9236 ← 27, 28, 36 [4 ] Rajendra BHATIA, MATRIX Analysis, Graduate Texts in Mathematics, Vol. 169, Springer-Verlag, New York, 1997 ← 11 [5] å NUMER Ical Methods in Matrix Calculations, Springer, Heidelberg, 2015. ← 10, 11 [6] C. Boutsidis, P. DRINEAS AND MAGDON-Immail, Near-Optimal Column-Based Matrix Reconstructure, Siam J. Comput. . 1137/ 12086755X. MR3504679 ← 40 [7] Petros Drineas, Ravi Kannan and Michael W. Mahoney, Fast Monte Carl o-Algorithms for Matriss. Выч. Вычоранeugовапапацацаты, 36 (2006), NO. 158-183, DOI 10. 137/S009753970442696. ] Petros Drineas, ILSE C. F. IPSEN, Eugenia-Maria Kontopoulou and Malik Magdon-Immail, Structure Convergence Results for Approximation of Dominant Subspaces of Block Krylov Spaces, Siam J. Matrix Anal. 7/1

[9] Petros Drineas, Ravi Kannan and Michael W. Mahoney, Fast Monte Carl o-algorithms for matrices. I. апрsyCoscирющее множение матриц, Siam J. Comput. 36 (2006), no. 1, 132-157, doi 10. 1137/s0097539704442684. MR2231643 ← 18, 20, 21, 24 [10] P. Drineas and M. W. Mahoney, Randnla: randomised numerical linear algebra, Communications of the ACM 59 (2016), no. 6, 80-90. ← 3 [11] P. Drineas, M. W. Mahoney, S. Muthukrishnan and T. Sarlós, Faster least squares approximation, numer. Maths. 117 (2011), № 2, 219-249, doi 10. MR2754850 ← 29, 36 [12] John C. Duchi, Introductory lectures on stochastic optimisation, The Mathematics of Data, IAS/Park City Math. (12)ジョン・C・ドゥチ，確率最適化入門講座，データの数学，IAS/Park City Math. Maths. Soc., Providence, RI, 2018. ← 3 [13] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., Johns Hopkins Studies in the Mathematical Sciences, Johns Hopkins University Press, Baltimore, MD, 1996. MR1417720 ← 4 , 11 [14] N. Halko, P. G. Martinsson and J. A. Tropp, Finding Structure with Randomness: Probabilistic algorithms for constructing approximate matrix decomposition, SIAM Rev. 53 (2011), no. 2, 217-288, doi 10. MR2806637 ← 37 [15] Wassily Hoeffding, Probability inequalities for sums of bounded random variables, J. Amer. Statist. 58 (1963), 13-30. MR0144363 ← 32 [16] John T. Holodnak and Ilse C. Ipsen, randomised random variables. F. Ipsen, Randomised approach to the grammar matrix: exact computation and probabilistic bounds, Siam J. Matr.

Petros Dorinius, Michael W. Mahony

10. 1090/PCMS/025/02 IAS/Park City Mathematics Series Volume 25, Pages 49-97 https: // doi. Org/10. 1090/PCMS/025/00830

Optimization algorithm for data analysis of Steven J.

Input 1. 1 Omitted 1. 2 Data Analysis Task Optimization 2. 3 End of 3r d-3r d-3rd quadrings 2. 5 Distribution evaluation of revolutionary c o-diversification Speech vector 2. 10 Logistic regression 2. 11 List of preparation information 3. 2. 2. 2 Enjoy and subtrallal 3. 3. 4 smooth functions of smooth functions 3. 6 speed 4. 1 Cool descent 4. 6 Relative gradient method<φ(xk )>50 51 51 52 52 54 54 56 56 57 57 60 61 64 64 65 67 67 71 72 72 72 74 75 75 75

Arithmetic subject classification 2010. Beginner class 14dxx; Intermediate 14DXX. Major text and chirard. Mathematical University Park City. © 2018 Steven J. Wright

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

Professional Slope Method 6. 3 NESTEROV acceleration gradient: For weak fingers 6. 4 Acceleration gradient Nes Terov: For tight convex 6. 5 Newton coefficient 7.

77 80 80 81 84 87 87 88 88 90 91 95 95

1. First, this notebook considers how to close the optimization task. Consider one of these orthodox formulas (1. 0. 1)<φ(xk )>→ R contains at least a continuous slope of the lips. If necessary, the auxiliary route on F is introduced, and these are introduced as the unevenness and continuity of Gessian Lip Shitsa. Other formulas we think --rn

Here, F is a function, as shown (1. 0. 1), ψ: RN → R is a function, usually convex, normally no n-smooth, and λ 0 is a regularized parameter. 1 We call (1. 0. 2) a regular minimization problem, but this is more desirable in the context of the application, with a specification of a certain structural nature in the solution by having a section containing ψ. , Or because it is a plausible solution. In the case of a convex function, the repetitive algorithm that generates a point of K = 0, 1, 2 ... (We, we give motivation to study problems in format, such as (1. 0. 1) and (1. 0. 2), from the universality in the application of data analysis. Therefore, in section 2 This is a typical data mining problem and the formula of the optimization problem, and the algorithms are described in a preliminary discussion in Section 4. The αZ + ( 1-α) Z∈S is established for all points Z, Z∈S for the extension of these methods. In the case of a convex function φ: rn → R, all Z, Z [0, 1] in the (convex) area φ (αZ + ( 1-α)). Z) When αφ (Z) + ( 1-α) φ (Z) is established, it is convex.

Stephen J. Wright < SPAN> Here, as (1. 0. 1), ψ: RN → R is a function, usually convex, normally no n-smooth, λ 0 is a regularized parameter. 1 We call (1. 0. 2) a regular minimization problem, but this is more desirable in the context of the application, with a specification of a certain structural nature in the solution by having a section containing ψ. , Or because it is a plausible solution. In the case of a convex function, the repetitive algorithm that generates a point of K = 0, 1, 2 ... (We, we give motivation to study problems in format, such as (1. 0. 1) and (1. 0. 2), from the universality in the application of data analysis. Therefore, in section 2 This is a typical data mining problem and the formula of the optimization problem, and the algorithms are described in a preliminary discussion in Section 4. The αZ + ( 1-α) Z∈S is established for all points Z, Z∈S for the extension of these methods. In the case of a convex function φ: rn → R, all Z, Z [0, 1] in the (convex) area φ (αZ + ( 1-α)). Z) When αφ (Z) + ( 1-α) φ (Z) is established, it is convex.

Stephen J. Light In here, F is a function, ψ: RN → R is a function, usually convex, normally no n-smooth, and λ 0 is a regularized parameter. 1 We call (1. 0. 2) a regular minimization problem, but this is more desirable in the context of the application, with a specification of a certain structural nature in the solution by having a section containing ψ. , Or because it is a plausible solution. In the case of a convex function, the repetitive algorithm that generates a point of K = 0, 1, 2 ... (We, we give motivation to study problems in format, such as (1. 0. 1) and (1. 0. 2), from the universality in the application of data analysis. Therefore, in section 2 This is a typical data mining problem and the formula of the optimization problem, and the algorithms are described in a preliminary discussion in Section 4. The αZ + ( 1-α) Z∈S is established for all points Z, Z∈S for the extension of these methods. In the case of a convex function φ: rn → R, all Z, Z [0, 1] in the (convex) area φ (αZ + ( 1-α)). Z) When αφ (Z) + ( 1-α) φ (Z) is established, it is convex.

Petros Dorinius, Michael W. Mahony

Section 5 describes the case (1. 0. 2) adjustment task. Section 6 describes the acceleration gradient method that achieves the highest difficulty level when the stability is worse than the basic gradient method. Section 7 describes the options that define the Newton Law and converge to the point that almost meets the conditions near the local minimum due to the smooth no n-convex functions. 1. 1. 1. 1. Our purpose is to briefly explain the more important algorizmuttles for smooth nonlinear optimization and adjustment optimization, and one of the main concepts of convergence Vo. 。 (In any context, "smoothness" means that the function is different by the number of times required, as the consideration is meaningful). In most cases, it is very easy to incorporate this doctrine here completely. In many other cases, it shows a link to a book that can find a complete confirmation. However, since we do not allo w-Property to the adjustment member B (1. 0. 2), this section does not consider clear appearance in the subtraction method or the mirror descending method. In addition, the stochastic slope method (class of a central position in modern machine learning) is not defined. All of these are described in the case of (1. 0. 2) adjustment tasks in all

The line is Roman (X, V, U and E. G. Further) in lowercase letters, and the vector is Roman (X, V, U, E. G.). (Actor with the actor is assumed to be attenuated vector). Transportation is classified by the upper structure index "T". The components of the matrix and vector are classified by the adaptive index, for example, AIJ or XJ. The repetition number is classified by the upper structure symbol, for example, XK. A large number of substances indicate that, for example, that RN actually means the Euclid space of the dimension N. Many symmetric material matrix N × N is classified by SRN × N.

Optimization-algorithm for data analysis

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

2. The optimization of data analysis tasks formula This section is briefly described in this section, which is a typical task of data analysis and machine learning, focusing on the formula as an optimization task. This list is by no means comprehensive. In many cases, there are various techniques for building a task as an optimization problem. I do not intend to give an overview of all of them. However, this list is an idea of help between data analysis and optimization. 2. 1. Practical datasets are often very confused. Data may be incorrectly marked, noise, incomplete, or ruined in other ways. Most of the difficult data analysis is traced by those who have to p-notch applications that do not change the important properties they want to detect as a result of analysis, "cleaning" the data, that is, the data. I will. DAZA and Johnson [19] states that "80 % of data analysis is spent on the process of data cleaning and preparation." We do not define the nuances of this process, focus on which part of the data-analysis assembly flow forms an inconsistency and forms an inconsistency. The dataset in the conventional analysis work is composed of M objects.

Here, AJ is a vector of the symptoms (or a matrix), and YJ is a director or annotation. (Each vs. (AJ, YJ) contains the same value and format for all j = 1, 2, .... Most of the reflected processes are often called "training". In the downstream, this parameter The work of setting the data is performed by the conversion: "Invite the characteristics of φ, this characteristics are actually φ (AJ) ≒ yj, j = 1, 2. Definition of appropriate sentences and terms. If you come up with it, "I have the optimization task.

Petros Dorinius, Michael W. Mahony

Stephen J. Wright < SPAN> Here, AJ is a vector of symptoms (or a matrix), and YJ is a director or annotation. (Each vs. (AJ, YJ) contains the same value and format for all j = 1, 2, .... Most of the reflected processes are often called "training". In the downstream, this parameter The work of setting the data is performed by the conversion: "Invite the characteristics of φ, this characteristics are actually φ (AJ) ≒ yj, j = 1, 2. Definition of appropriate sentences and terms. If you come up with it, "I have the optimization task.

Here, the J-Dee member (AJ, YJ; x) is a scale of contradiction between φ (AJ) and YJ, and X is a characteristic vector that defines φ.

Stephen J. Wright Here, AJ is a vector of symptoms (or a matrix), YJ is a director or annotation. (Each pair (AJ, yj) contains the same value and format for all j = 1, 2, .... Most of the reflection process is often called "training". In the downstream, this parameter The work of setting the data is performed by the conversion: "Invite the characteristics of φ, this characteristics are actually φ (AJ) ≒ yj, j = 1, 2. Definition of appropriate sentences and terms. If you come up with it, "I have the optimization task.

Here, the J-Dee member (AJ, YJ; x) is a scale of contradiction between φ (AJ) and YJ, and X is a characteristic vector that defines φ.

Stephen J.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

And we hope that the function φ will work perfectly on, for example, unnecessary data points and observed subset D. For this purpose, the formula of the optimization can be modified by several methods: penalties that limit some scale of the connection restriction or complexity function (this method is generally a method. Original or regulator). In the latter stage of the optimization process, the leader generates overkills, so there is another consistency in the injury end of the optimization method.

Optimizatio n-algorithm for data analysis

2. 2. Here is the data point (AJ, YJ) of RN x R (AJ, YJ), 1 T 1 A X-Y22, (AJ X-YJ) 2 = 2m 2m M. < SPAN> And we hope that the function φ will work perfectly on, for example, unnecessary data points and observed subset D. For this purpose, the formula of the optimization can be modified by several methods: penalties that limit some scale of the connection restriction or complexity function (this method is generally a method. Original or regulator). In the latter stage of the optimization process, the leader generates overkills, so there is another consistency in the injury end of the optimization method.

Optimizatio n-algorithm for data analysis

Petros Dorinius, Michael W. Mahony

Optimizatio n-algorithm for data analysis

2. 2. Here is the data point (AJ, YJ) of RN x R (AJ, YJ), 1 T 1 A X-Y22, (AJ X-YJ) 2 = 2m 2m M.

Here, A is a straight line with ATJ, J = 1, 2, ... In the above term, the function φ is determined to be φ (a): = at x (we also add additional parameter β∈R and determine φ (a): = at x + β. By doing so, the unatulated cut can be introduced). This formalization is accurate for YJ's object, but when there is I. I. D. Gaussa noise, it is statistically motivated as the evaluation of the maximum reliability of X. The method of randomized linear pending for larg e-scale examples of this task is discussed in the fifth section of DRINEAS and MAHONI [20]. Some corrections (2. 2. 1) impose a desirable structure in X as a result. For example, tikhono v-REGULATED with a QUADRATIC 2-norm, Which is 1 A X-Y22 + λx22 With less sensitivity to perturments in the data ( AJ, YJ). In principle, Lasso 1 A X-Y22 + λx1 (2. 2. 2) min X 2M formula gives a relatively small solution, that is, unit equal components. [42] This formalization clarifies the position of the non-similar components in X: the position of the non-similar components in x clarifies the components of AJ, which plays an important role in determining YJ's object. In addition to the statistical charm, the prediction that depends on a small number is potentially simple.

Here A, B: = (above B) trace. Here we can think of AJ as "exploring" unknown matrix X.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

Here A, B: = (above B) trace. Here we can think of AJ as "exploring" unknown matrix X.

Petros Dorinius, Michael W. Mahony

Here A, B: = (above B) trace. Here we can think of AJ as "exploring" unknown matrix X.

Stephen J. Wright

Configuration (when the components of AJ are selected from the distribution) or one element study (when 1 is included in a space with AJ, and 0 in other places). The regulatory version (2. 3. 1) that guides lo w-ranking inference X includes image M 1 (AJ, X-YJ) 2 + λ x *, (2. 3. 2) MIN X 2m J = 1.

Here, X * is a nuclear norm, which gives the required amount of the specific value of X. [39] The nuclear norm plays the same role as Norm 1 V (2. 2. 2). However, nuclear norm is regarded as a kind of difficult function, for example, a convex, for example, as the formula of (2. 3. 2) is still convex. This formalization is a real X-ranked rank, and the A J-SeaRch matrix satisfies the nature of the "restricted quantity" in which the random matrix is usually satisfied, but does not satisfy the matrix with one no n-roasted component. It is possible to show that statistically abundant inference is actually distinguished. In this formula, the current format is a parakeet healer (in rough terms, there is no more important ingredients than others), and A J-Investigation is considered to be composed of a single ingredient. Is fair. [10] Here, L is RN x R, R is RP x R, but R (n, P) is drawn from R. X = LRT B (2. 3. 1), and solve 1 (AJ, LR T-YJ) on 2. 2m m. < SPAN> Configuration (if the components of AJ are selected from the distribution) or one element study (when AJ contains 1 in a space with AJ and 0 in other places). The regulatory version (2. 3. 1) that guides lo w-ranking inference X includes image M 1 (AJ, X-YJ) 2 + λ x *, (2. 3. 2) MIN X 2m J = 1.

Here, X * is a nuclear norm, which gives the required amount of the specific value of X. [39] The nuclear norm plays the same role as Norm 1 V (2. 2. 2). However, nuclear norm is regarded as a kind of difficult function, for example, a convex, for example, as the formula of (2. 3. 2) is still convex. This formalization is a real X-ranked rank, and the A J-SeaRch matrix satisfies the nature of the "restricted quantity" in which the random matrix is usually satisfied, but does not satisfy the matrix with one no n-roasted component. It is possible to show that statistically abundant inference is actually distinguished. In this formula, the current format is a parakeet healer (in rough terms, there is no more important ingredients than others), and A J-Investigation is considered to be composed of a single ingredient. Is fair. [10] Here, L is RN x R, R is RP x R, but R (n, P) is drawn from R. X = LRT B (2. 3. 1), and solve 1 (AJ, LR T-YJ) on 2. 2m m. Configuration (when the components of AJ are selected from the distribution) or one element study (when 1 is included in a space with AJ, and 0 in other places). The regulatory version (2. 3. 1) that guides lo w-ranking inference X includes image M 1 (AJ, X-YJ) 2 + λ x *, (2. 3. 2) MIN X 2m J = 1.

Here, X * is a nuclear norm, which gives the required amount of the specific value of X. [39] The nuclear norm plays the same role as Norm 1 V (2. 2. 2). However, nuclear norms are regarded as a kind of difficult function, and, for example, the formula of (2. 3. 2) is considered to be still convex. This formalization contains a lo w-ranking real number, and the A J-SeaRch matrix satisfies the nature of the "restricted quantity" in which the random matrix is usually satisfied, but does not satisfy the matrix with one no n-roasted component. It is possible to show that statistically abundant inference is actually distinguished. In this formula, the current format is a parakeet healer (in rough terms, there is no different ingredients in it), and A J-Investigation is considered to be composed of a single ingredient. Is fair. [10] Here, L is RN x R, R is RP x R, but R (n, P) is drawn from R. X = LRT B (2. 3. 1), and solve 1 (AJ, LR T-YJ) on 2. 2m m.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

Actually L 0, R 0.

Optimizatio n-algorithm for data analysis

2. 5. In this task, the Y J-marker is considered zero, and the vector aj∈RN is considered an autonomous study of random vector a∈RN with a zero average. The partial coordinated matrix built for these studies includes the image M 1 s = aj ATJ. M-1 J = 1

The SIL element is the evaluation of coexistence between the I-M substance of the vector A of any value and the L-M substance. Our focus is to calculate the evaluation of the X-Revolutionary Co-Diversification Academic, which is considered sparse. In particular, if it is XIL = 0, it can be concluded that the component A of I-I and L-I is significantly autonomous. (In other words, if the meaning of other N-2 components A) is common, they are autonomous). In other words, X's no n-zero space means an arc in a dependencies, and its node satisfies N components A. )

S, X-log Det (x) + λx1, < SPAN> In this formula, rank RAN is "strictly incorporated" in the definition of X, and as a result, there is no need to connect regulatory members. This formalization is much more compact than (2. 3. 2), and the components of (L, R) are much less than NP, as in (n+p) r. This defect deserves attention. The intensive purpose of advanced research, which is started in 9] and is still based on statistical information, is considered to be benign in many settings, and in fact data (AJ, YJ). By specific specific guesses, it is possible to derive a good conclusion from the formula (2. 3. 3) by the strict choice of j = 1, 2, ... m and algorithm strategies. The key to this good behavior is that this formula is not essential, but it is considered an approach to the solved issue in a sense: if there is absolute supervision X, the approach with rank R is the Lord. There is a probability of finding a decomposition method X according to the specific value of the left and right vectors and the definition of L and R. 2. 4. In factor decomposition of no n-contributed queues in computer vision, hemometry, document clustering, etc., the position of the moment L and R actually given in (2. 3. 3) is strongly required.

Petros Dorinius, Michael W. Mahony

Optimizatio n-algorithm for data analysis

2. 5. In this task, the Y J-marker is considered zero, and the vector aj∈RN is considered an autonomous study of random vector a∈RN with a zero average. The partial coordinated matrix built for these studies includes the image M 1 s = aj ATJ. M-1 J = 1

The SIL element is the evaluation of coexistence between the I-M substance of the vector A of any value and the L-M substance. Our focus is to calculate the evaluation of the X-Revolutionary Co-Diversification Academic, which is considered sparse. In particular, if it is XIL = 0, it can be concluded that the component A of I-I and L-I is significantly autonomous. (In other words, if the meaning of other N-2 components A) is common, they are autonomous). In other words, X's no n-zero space means an arc in a dependencies, and its node satisfies N components A. )

S, X-log Det (x) + λx1, in this formula, rank RAN is "strictly incorporated" in the definition of X, and as a result, there is no need to connect regulatory members. This formalization is much more compact than (2. 3. 2), and the components of (L, R) are much less than NP, as in (n+p) r. This defect deserves attention. The intensive purpose of advanced research, which is started in 9] and is still based on statistical information, is considered to be benign in many settings, and in fact data (AJ, YJ). By specific specific guesses, it is possible to derive a good conclusion from the formula (2. 3. 3) by the strict choice of j = 1, 2, ... m and algorithm strategies. The key to this good behavior is that this formula is not essential, but it is considered an approach to a solved problem: if there is absolute supervision X, the approach with rank R is the main. There is a probability of finding a decomposition method X according to the specific value of the left and right vectors and the definition of L and R. 2. 4. In factor decomposition of no n-contributed queues in computer vision, hemometry, document clustering, etc., the position of the moment L and R actually given in (2. 3. 3) is strongly required.

Actually L 0, R 0.

Optimizatio n-algorithm for data analysis

2. 5. In this task, the Y J-marker is considered zero, and the vector aj∈RN is considered an autonomous study of random vector a∈RN with a zero average. The partial coordinated matrix built for these studies includes the image M 1 s = aj ATJ. M-1 J = 1

The SIL element is the evaluation of coexistence between the I-M substance of the vector A of any value and the L-M substance. Our focus is to calculate the evaluation of the X-Revolutionary Co-Diversification Academic, which is considered sparse. In particular, if it is XIL = 0, it can be concluded that the component A of I-I and L-I is significantly autonomous. (In other words, if the meaning of other N-2 components A) is common, they are autonomous). In other words, X's no n-zero space means an arc in a dependencies, and its node satisfies N components A. )

S, X-log Det (x) + λx1,

Here, SRN × N is a large number of Nx N, and X 0 means that X is exactly determined, and it is X1: = n I, L = 1 | ([17, 25]. reference). The main components in this task are the same as the previous section: We have a partial coordinated matrix S that are evaluated by many basic random vectors. The main component of this matrix is a personal vector suitable for large personal values. In many cases, the intelligent intelligence of the sparse components gives the approximation of a major personal vector, but still does not contain enough no n-base. The obvious optimization formula of this task is Max VT SV SV S. T. V2 = 1, V0 K.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

S, M S. T. M 0, I, M = 1, M1 ρ,

Petros Dorinius, Michael W. Mahony

Stephen J.

So, the columns are orthogonal to each other, and have a non-base. The convex easing of this task is still a sem i-intellectual program and is (2. 6. 3).

S, M S. T. 0 M I, I, M = 1, M1 ρ.

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

¯ s. T. F2 1, f2. 1 r,

F2. 1: = [15]. The last adjustment section is often referred to as the regulation section of groupsperers or the regulation of glue prasso. (Initially introduced in the adjustment of 1 in similarity is described in [44]) 2. 7. The decomposition of the basal plus in the low rank matrix is another paradigm-The decomposition is faint. It is a partial or compassionate observation N × P matrix Y of the required amount of a compensation and low rank matrix. Complete observation task convex formatizatio n-min m ∗ + λs1 s. T. T. Y = m + s, n

Here, s1: = i = 1 j = 1 | [11, 14]. The formula of a smal l-sized indication of a roar in the director contains the correct one: 1 LRT + S-Y2F (completely observed) L, R, S 2 1 Min Part (LRT + S-- Y) 2F (partially observed), L, R 2 Here, φ gives the position of the observed record Y, and part is the projective of this large number [15, 48]. One of the applications of these formulas is a timid PCA, a lo w-ranking ratio gives important ingredients, and the sparse ratio is a survey with "discharge". Another usage is the distribution of the intention of the front and rear aspects when processing a video. Here, each column gives a pixel of the y in one frame video, and each line Y shows the time of the first pixel.

Petros Dorinius, Michael W. Mahony

The algorithms that determine X are described in [1], and each AJ is given a projection diagram of an avian dance that is gradually executed in order. That correctness

Zopt 22/2 (XUA) T Xua Zopt 22 = (Xua) T XB⊥22.

The relative key axis of a queue X, that is, a matrix X, does not need to have a certain number of components that are important for other things. The test of local convergence of this method is shown in [2]. 2. 9. Classification (SVM) by support vector (SVM) is considered a traditional paradigm of machine research. In this task, there is a group of (AJ, YJ) with aj∈Rn and yj∈ as input, and vector x∈RN and scalar β∈R are searched (2. 9. 1a).

Any pair (X, β) that satisfies these conditions defines the separation ultr a-plane of the RN that separates the case of "positive" and the "negative" case. (In the words of section 2. 1, function φ can be modified as φ (AJ) = Sign (atj X-β)). Among all the separation ultr a-planes, the minimum X2 is considered to be the maximum distance to the maximum margin between two classes, that is, the closest AJ of the class of the class. A problem that seeks a shared super plane can be built as an optimization problem, and this problem is defined in the section (2. 1. 2): 1 Max ( 1-YJ (atj x-β), 0) m m m m

Note that the J-number in this section is equal to zero if the conditions (2. 9. 1) are satisfied, and otherwise they are positive. H (x, β) = 0, including a group (x, β) that does not exist, is the optimal price (2. 9.). It is a very close group to satisfy 1). In (2. 9. 2), section λx22 is often added, but here is a small flattering parameter, which actually leads to the correct regular version: 1 Max ( 1-YJ (ATJ (ATJ x-β), 0) + λx22.

If λ is very small (although positive) and a distribution ultr a-plane exists, the pair (X, β) that minimizes (2. 9. 3) is considered a distribution ultr a-plane with the largest margin. The characteristics of the margin correspond to the purpose of generalization and robustness. For example, if the observation data (AJ, YJ) is extracted from the basic "clouds" consisting of a positive and negative case, the inference based on the margin limit is usually extracted from the same "cloud". Nearly distinguish the set. The minimization problem (2. 9. 3) is a convex secondary plan-convex secondary target and linear constraine agreement by the method of introducing the variable SJ, J = 1, 2, ... You can write. moreover

Stephen J.

Sketch 2. 9. 4. Linear classification using a support vector machine (one class is round, another class is expressed in square). One of the possible deformation in the distribution ultr a-plane is shown on the left. If the observed data is considered empirically selected from the clouds at the basic data point, this plane cannot separate two clouds (center). The ultr a-plane, which separates the largest boundary, works better than any other ultr a-plane (right side).

1 T 1 S + λx22, 2 X, β, S M Min

SJ 1-YJ (ATJ X-β),

Where ∈ 1 = often occurs, it is impossible to find a hyper plaster that is completely distinguished from flat and negative cases, which are required as a classification. One conclusion is that all initial AJ data vectors, which have the highest dimensions or more in Euclid, are reconstructed, then vector (AJ), j = 1, 2,. Conditions (2. 9. 1) are (1, 1,., 1) t

ζ (AJ) x-β -1 with yj = -1,

This is connected to the next analog (2. 9. 3): 1 1 Max ( 1-YJ (AJ) T X-β), 0) + λx22. M 2 m

With a revolutionary conversion of RM, the plane is no n-linear, probably incoherent, and is considered to be much closer to a massive classifier than (2. 9. 3), which is often obtained from (2. 9. 3). For example, (2. 9. 7) can be literally designed as a convex secondary plan, as led (2. 9. 3) (2. 9. 5). If you take the binity of this secondary program, you can get another convex secondary program of M variables: 1 (2. 9. 8) MINM αT Q α-1t ζ (AK) T ζ (AL) ,,

y = (y1, y2, ..., ym) t,

1 = (1, 1,., 1) t ∈ rm.

Optimization algorithm for data analysis

Interestingly, the task of (2. 9. 8) can actually be settled and solved without the clear knowledge or decision of reflection. All you need is a technology that determines the component of Q. This can be organized by the support of the function of kernel K: RN × RN → R, and K (AK, al) is replaced with ζ (AK) T ζ (al) [4, 16]. For example, this is the "core trick". (Kernel K function may be used to build classification function φ from the conclusion (2. 9. 8). Gaussian kernel is considered a wel l-known set of kernel: k (ak, al): = EXP (- a-a c-Al 2 /(2σ)), here is a flattery parameter. 2. 10. Logistic regression regression can be regarded as a variant of binary systematization that supports the support vector machine, and in return, verification of classes with systematization functions φ. As distinct from an impossible distribution, it returns the probability of A matching a class or another class. We are looking for a "probability function" with vector x∈RN as parameters.

p (a; x): = (1 + exp (on x)) --1,

SJ 1-YJ (ATJ X-β),

p (aj; x) ≈ 1, when yj = +1;

p (aj; x) ≈ 0, when yj = -1.

(Pay attention to monotony C (2. 9. 1)). By maximizing the credit function of credibility, it is possible to find an appropriate meaning X: ¡ 1 £ ( 1-p (aj; x)) + log P (AJ; x) ¦. (2. 10. 3) l (x): = m j: yj = -1

By introducing the λx 1-regulator as shown below, the code according to this model can be satisfied: ⎡ (2. 10. 4) max log ( 1-p (aj; x)) + log p ( X) ⎦x1, x M J: yj = -1

Here, λ & amp; amp; gt; 0-regulatory parameters. (Note that this task is settled as maximizing it, but not to subtract the members of λx 1-regulatory parameters and not add them to the target). As you can see later, this member led a decision that some of the ingredients of X were unmatched, and knew only those components corresponding to the no n-similar value of X, p (a; x). Can be evaluated. An important extension of this method is a mult i-class (or polymorph) logistic regression that AJ Data Vector belongs to two or more classes. Such applications are often seen in modern data analysis. For example, in a speech recognition system, the M-class can be one of thousands of different basic sounds, the potential, or the potential to be vocalized.

Stephen J.

A few tens of milliseconds. The multidrawal logistic regression task requires a different function of the PK probability of each class K∈. These functions are determined as the following: (2. 10. 10. 5)

EXP (ON X [k]), pk (a; x): = m t L = 1 EXP (a x [l])

Here, x: =. Note that all A, all k = 1, 2, ..., m, pk (a) ∈ (0, 1), and mk = 1 pk (a) = 1. In the multi-class logistic regression task of YJ-Marking, these are the vector of RM, the elements are determined as follows: when AJ belongs to the class K (2. 10. 6) YJK = 0. Similarly (2. 10. 2), we aim to determine the vector x [k] that will be (2. 10. 7a).

Finding the values of X [k] that satisfy these conditions can be r e-designated as a task that maximizes the reliability of the number: M M 1 T T YJ (x AJ) --log EXP (x [] AJ) (2. 10. 8) l (x): = m j = 1

In this formula, it is possible to connect the regulatory conditions of the group sparse to select the symptom set of AJ vector, which is integrated for each class that effectively distinguishes classes. 2. 11. Thorough research on the deepest neural network is that once, in many cases, a logistic regression with multiple classes, that is, a highly possible M class to systematize the databector. Developed to execute such functions, M2 is large in some main applications. The difference is that the actor of the data from A receives several structural conversions, as described in the last section, which will be overlooked by the similarity of multipl e-class logistic regression. Figure 2. 11. 1 The normal neural network shows the main concept. In this figure, the AJ databector is in the lower part of the network, and each node in the lower layer is compatible with one component AJ. After this, the vector moves across the network and receives a structured nonlinear reconfiguration when the first layer is switched. The normal reconstruction of the L-1 layer for the input vector Alj on the LAS is Alj = σ (W L A l-1 J + GL), Alj = σ (W L A l-1 J + GL), Alj = σ (W L) A l-1 J + GL), Alj = σ (W L A l-1 J + GL).

Optimizatio n-algorithm for data analysis

Figure 2. 11. Here, W L is a dimensional matrix | × | AL-1 J |, G is a vector of length | AJ |, σ is a component non-linear reconstruction, and D is the lowest and top layer. It is the number of shelter layers defined as strict layers in between. Figure 2. Figure 2. If A0J is recognized as a "raw" input vector AJ and AD J is a vector, FIG. There is the same as R: < SPAN> In this formula, connect the regulatory conditions of the group sparse to select the symptom set of AJ vector, which is aggregated for each class that effectively distinguishes classes. Is possible. 2. 11. Thorough research on the deepest neural network is that once, in many cases, a logistic regression with multiple classes, that is, a highly possible M class to systematize the databector. Developed to execute such functions, M2 is large in some main applications. The difference is that the actor of the data from A receives several structural conversions, as described in the last section, which will be overlooked by the similarity of multipl e-class logistic regression. Figure 2. 11. 1 The normal neural network shows the main concept. In this figure, the AJ databector is in the lower part of the network, and each node in the lower layer is compatible with one component AJ. After this, the vector moves across the network and receives a structured nonlinear reconfiguration when the first layer is switched. The normal reconstruction of the L-1 layer for the input vector Alj on the LAS is Alj = σ (W L A l-1 J + GL), Alj = σ (W L A l-1 J + GL), Alj = σ (W L) A l-1 J + GL), Alj = σ (W L A l-1 J + GL).

Optimizatio n-algorithm for data analysis<(x, t) ∈ Rn × R : v, x + at c>Figure 2. 11. Here, W L is a dimensional matrix | × | AL-1 J |, G is a vector of length | AJ |, σ is a component non-linear reconstruction, and D is the lowest and top layer. It is the number of shelter layers defined as strict layers in between. Figure 2. Figure 2. If A0J is recognized as a "raw" input vector AJ and AD J is a vector, FIG. There is the same as R: In this formula, it is possible to connect the regulation conditions of the group sparse to select the symptom set of AJ vector, which is integrated for each class that effectively distinguishes the class. be. 2. 11. Thorough research on the deepest neural network is that once, in many cases, a logistic regression with multiple classes, that is, a highly possible M class to systematize the databector. Developed to execute such functions, M2 is large in some main applications. The difference is that the actor of the data from A receives several structural conversions, as described in the last section, which will be overlooked by the similarity of multipl e-class logistic regression. Figure 2. 11. 1 The normal neural network shows the main concept. In this figure, the AJ databector is in the lower part of the network, and each node in the lower layer is compatible with one component AJ. After this, the vector moves across the network and receives a structured nonlinear reconfiguration when the first layer is switched. The normal reconstruction of the L-1 layer for the input vector Alj on the LAS is Alj = σ (W L A l-1 J + GL), Alj = σ (W L A l-1 J + GL), Alj = σ (W L) A l-1 J + GL), Alj = σ (W L A l-1 J + GL).

Optimizatio n-algorithm for data analysis

Figure 2. 11. Here, W L is a dimensional matrix | × | AL-1 J |, G is a vector of length | AJ |, σ is a component non-linear reconstruction, and D is the lowest and top layer. It is the number of shelter layers defined as strict layers in between. Figure 2. Figure 2. If A0J is recognized as a "raw" input vector AJ and AD J is a vector, FIG. There is the same as R:

-Logistic function: T → 1/(1 + e-t); -loop loss: T → Max (T, 0); -Bernouy: 1/(1 + e-t) gives 1, 0 in an unpleasant case Random function to give. Each node of the upper layer corresponds to a specific class, and the output of the node is the probability that the input vector matches each class. As already mentioned, Softmax operators are usually used to reconstruct the input vector that has been reborn in the second upper layer (D layer) into a set of coefficients. Each input vector AJ specifically, the YJK marker is associated as (2. 10. 6), indicating which AJ belongs to the M class. The parameters of this neural network are the matrix-vector vs (WL, gll), L = 1, 2, ... D, which is converted to the input vector AJ into its adhesive at the top hidden layer. do. In addition, it is a feature of X-processed class logistic regression executed at the top line, and here X is literally, for example,, for example, ..., for example,, for example, X. ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ... ,,,,,,,,,,,,,,,, We want to select all of these features in this way so that the network is completely controlled by the systematization of training data. Use the name W to convert the asylum layer, that is, (2. 11. 2)

W: = (w 1, g1, w 2, g2, ..., w d, gd),

Stephen J.

Determined, like X: = 2. 10, a function of loss for thorough research can be exported by a clock: m m 1 yj (xt [xt [] ad. (2. 11) (2. 11) 3) l (w, x): =): = j (w)) -log J (W)) M J = 1 < Span> -Logistic function: T → 1/(1 + e-t); - Loop loss: T → Max (T, 0); -Bernouy: 1/(1 + e-t) gives 1 with the possibility of 1/(1 + e-t), and in unpleasant cases, random functions are given 0. Each node of the upper layer corresponds to a specific class, and the output of the node is the probability that the input vector matches each class. As already mentioned, Softmax operators are usually used to reconstruct the input vector that has been reborn in the second upper layer (D layer) into a set of coefficients. Each input vector AJ specifically, the YJK marker is associated as (2. 10. 6), indicating which AJ belongs to the M class. The parameters of this neural network are the matrix-vector vs (WL, gll), L = 1, 2, ... D, which is converted to the input vector AJ into its adhesive at the top hidden layer. do. In addition, it is a feature of X-processed class logistic regression executed at the top line, and here X is literally, for example,, for example, ..., for example,, for example, X. ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ... ,,,,,,,,,,,,,,,, We want to select all of these features in this way so that the network is completely controlled by the systematization of training data. Use the name W to convert the asylum layer, that is, (2. 11. 2)

W: = (w 1, g1, w 2, g2, ..., w d, gd),

Stephen J.

Determined, like X: = 2. 10, a function of loss for thorough research can be exported by a clock: m m 1 yj (xt [xt [] ad. (2. 11) (2. 11) 3) l (w, x): =): = j (w)) -log J (W)) M J = 1-Logistic function: T → 1/(1 + e-t); -Loop loss: T → MAX (T, 0); -Bernouy: 1/(1 + e-t) gives 1 in the possibility of 1/(1 + e-t), and is a random function that gives 0 in unpleasant cases. Each node of the upper layer corresponds to a specific class, and the output of the node is the probability that the input vector matches each class. As already mentioned, Softmax operators are usually used to reconstruct the input vector that has been reborn in the second upper layer (D layer) into a set of coefficients. Each input vector AJ specifically, the YJK marker is associated as (2. 10. 6), indicating which AJ belongs to the M class. The parameters of this neural network are the matrix-vector vs (WL, gll), L = 1, 2, ... D, which is converted to the input vector AJ into its adhesive at the top hidden layer. do. In addition, it is a feature of X-processed class logistic regression executed at the top line, and here X is literally, for example,, for example, ..., for example,, for example, X. ,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, ... ,,,,,,,,,,,,,,,, We want to select all of these features in this way so that the network is completely controlled by the systematization of training data. Use the name W to convert the asylum layer, that is, (2. 11. 2)

W: = (w 1, g1, w 2, g2, ..., w d, gd),

Stephen J.

Determined, like X: = 2. 10, a function of loss for thorough research can be exported by a clock: m m 1 yj (xt [xt [] ad. (2. 11) (2. 11) 3) l (w, x): =): = j (w)) --log J (W)) m j = 1

In fact, this is the same as the function (2. 10. 8) used in the output of the top DD of the coating layer AD J (W). We write AJ (W) to solve the clear AJ from the feature from (2. 11. 2) from (2. 11. 2) and from the input vector AJ. (We have a mult i-class logistic regression (2. 10. 8), for example, a shelter like, for example, D = 0, W-0, AD J = AJ, J = 1, 2. = 1. It can be considered as a personal case of a thorough research. The second reason is that it is very difficult to find a minimum mass of mass (2. 10. 8), which is used in the output of the top D D (W). It is the same as (2. 11. 2), and to solve AJ from the input vector AJ (we). Regarding (2. 10. 8), for example, d = 0, w --0, AD j = aj, j = 1, 2. = 1, personal research in thorough research. First of all, you can think of it, probably, the most important thing is to find the "Landscape". The second reason is that this is the same as the output of the top D D (W). Features from 11. 2) Write AJ (W) (2. 10. 8) to solve AJ from the input vector AJ. In fact, it can be considered as a personal case of thorough research without shelters like D = 0, AD J = AJ, J = 1, 2. = 1. The most important thing is that it is very difficult to find the "Landscape" L, which is not a mixture of W. On

3. Preliminary conditions In this case, some conditions are based on the subsequent section analysis. These include the necessary pr e-conditions related to the smooth convex function and the no n-smooth convex functions, some conditions of the Taylor, some conditions, the optimization conditions, and the proximity effect. In this section, it is assumed that F is a reflection from RN to R∪ and is continuous in its own valid area D: =.

The large amount of convex ω ω RN has the following properties:

X, y ∈ ω ⇒ ( 1-α) x + αy ∈ ∈ ω ω ω ω [0, 1].

This note handles a large amount of closed convex. For convex large capacity ω ⊂ RN, we recognize the indicator function IΩ (x) as follows: 0, x ∈ ω ω ω (x) = +∞ Otherwise. Index functions are useful when deriving optimal standards for restricted conditions, such as algorithm development. Equipment with constraints (3. 2. 2)

Can be equally renewed as follows: (3. 2. 3)

SJ 1-YJ (ATJ X-β),

Stephen J. Wright

Definition 3. Vector v∈Rn is considered to be under the slope of f (x + d) f (x) + VT D in point X.

The part of the fixing valley given by F (x) is a number of partial validations in all partial valley F in point x. Partial differentials satisfy the monotonous properties. Theorem 3. 2. 7. 7. a ∈f (x) and b ∈f (y), THEN ( a-b) t ( x-y) 0. Prove. From the bulge F and the definition A, B, F (Y) F (X) F (x) + ON ( Y-X) and F (X) F (Y) + Bt ( X-Y) are emitted. This result follows the addition of these two inequality. The minimum value can be easily characterized from the viewpoint of the differential valley. Theorem 3. 2. 8. The point x * is the minimization of convex focus F only when 0 ∈ ∂f (x *). If x = x ∗ and v = 0 are substituted in the proof definition 3. 2. 6 and 0 ∈ ∂f (x ∗), f (x ∗ + d) f (x ∗) for all d∈Rn. You can get it. SUBDIFFERIOP is a generalized concept of a non-core convex function. If the theorem 3. If f is different and different from a point X, ∂f (x) =. Conversely results will also occur. In particular, if the unevenly differential function F has the only deviation slope in point X, F is different in a gradient (see [40, THEOREM 25. 1]). 3. 3. Taylor Theorem Taylor THEOREM is a basic theorem for optimizing smooth nonlinear functions.

When F is continuously differentiated, 1 (3. 3. 4) ∇f (x + p) = ∇f (x) + ∇2 f (x + tip) p Daw, 0.

1 f (x + p) = f (x) + ∇f (x) t p + pt ∇2 f (x + tip) p, 2

In the case of any lection∈ (0, 1).

Optimization algorithm for data analysis

When F is continuously differentiated by a constant L and lips, you can bring out the important consequences of this theorem.

About all x, y∈Rn.

1 F (Y) - F (X) - ∇f (X) - ∇f (Y) - ∇f (Y) - ∇f (Y) - ∇f (X) by submitting y = x + p to (3. 3. 2) and drawing members ∇f (x) t ( y-x) from both sides. X) t ( y-x) = [∇f (x + perf ( Y-x)) -∇f (x)] T ( y-x) Dival. 0.

(3. 3. 6) can obtain [∇f (x + perf ( y-x)) -∇f (x)] t ( y-x) ∇f (x + perf ( Y-x)) 。 -∇f (x) y-x Lisin g-x2. If this area is replaced with the previous integral, L (3. 3. 7) f (y) -F (x) -∇f (x) t ( y-x) Y-x2 will be obtained. 2 In Section 3 or later, ass shall be continuously differentiated and convex. (3. 3. 8) from the definition of valle (3. 2. 4) and ∂f (x) =.

f (y) f (x) + ∇f (x) t ( y-x),

About all X, y∈RN.

(3. 2. 5), defined a "strong bulge with module γ". When f is differented, the following equivalent definitions were obtained by permittance (3. 2. 5) and α ↓ 0. γ (3. 3. 9) f (y) f (x) + ∇f (x) t ( y-x) + y-x2. 2 The result is obtained. Theorem 3. 3. 10. (3. 2. 5) is satisfied, and when ∇f is a constant L and the lips are evenly continuously console F, the γ L Y-X2 F (Y) - F for any X, Y, Y γ L Y-X2 F (Y) - F (X) -∇f (x) t (x) t (x) t (x) t (x) t (x) t (x) t (x) t (x) t (x) t (x) t (x) y-x2 。 (3. 3. 11) 2 2 2 2 2 For more convenience, the condition number κ is determined as follows: l (3. 3. 12) κ: =. Γога f дважды ныво да, ыожараракport на на на ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а ∇а на ∇а на на на на на акакакакакакакакакакакакакакакакакажажажажаж X). Especially (3. 3. 11) can be equally (3. 3. 13).

When F is strictly convex and has a secondary function, κ is defined in (3. 3. 12) and is a gemnent gessian in the sense of normal linear algebra. The very convex functions have the only minimum chimch, as shown in the future. Theorem 3. If F is possible and strongly convex γ & amp; amp; amp; g t-module; 0, there is the only one with minimal chicken x * f-function f.

Stephen J.

Proof. First, any point X0 is a restricted and restricted, that is, compact. Suppose that there is a next number such as X → φ, f (x) f (x0) for contradiction.

SJ 1-YJ (ATJ X-β),

1 1 1 1 2 (3. 5. 2) mλ, h (x): = infh h (u) + u-x = inf λh (u) + u-x. U 2λ λ u 2-function λh The child is infinite in (3. 5. 2), the meaning of u, that is,

1 (3. 5. 3) Proxλh (x): = arg minus λh (u) + u-x2. U 2 (3. 5. 3) 最 から (see 3. 5. 3), (see 3. 5.) 4)

0 ∈ ∂hh (Proxλh (x)) + (Proxλh (x) -x) appears.

The Moro's wrapping line can be regarded as a kind of smoothing or adjustment of a function H. Even if H accepts infinite values against any X∈RN, it has a final value for all X. In fact, it is distinguished from H (x) = (x) = ( x-prox λh (x)) with a gradient of 1 ∇mλ everywhere. Until λ and X * are minimized mλ, h, and only at that time. Nearly applied elements satisfy the nature of inaccuracy. X-Proxh (x) ∈ ∂ (PROX ∂ (X)) appears from the optimal conditions (3. 5. 4) in 2 points x and y

Y-Proxλh (y) ∈ ∂ ∂ (Proxλh (y)).

Using monopoly (Lemma 3. 2. 7) is as follows.

T (1/λ) ( x-proxλh (x)) - ( y-proxλh (y)) (proxλh (x) -PROXλh (y)) 0,

Stephen J.

SJ 1-YJ (ATJ X-β),

1 1 Proxorda (x) = arg minus λiΩ (u) + u-x2 = arg minus u-x2, u 2 u] 2-H (x) = x1. When the definition (3. 5. 3) is assigned, the minimum minimization is decomposed into n-individual components, and the first ingredient Proxλ-1 (x).

1 Proxλ-1 (x) = arg min λ | + (UI-xi) 2. 5. 5) [Proxλ-1 (x)] i = 0 if xi ∈ [-, λ]; + λ if x 0: (4. 5. 1)

∇f (XK) T D K-ON (XK) DK.

The length of the step length αK is permanent C1 and C2 defined in the permanent C1 and C2 in 0 0, and the permanent C1 and C2 in 0 0 under the PRO X-operator. Assuming that the following conditions (4. 5. 2) under the following are satisfied. By substituting this definition, it is possible to guarantee that XK + 1 is the solution of the φ closed task of (5. 0. 1). (5. 0. 3) xk + 1: = arg min ∇f (XK) t ( z-xk) + z 2ak 2 for

In analysis of this section, I was taken care of by L. Vandenberg's memo created from 2013 to 2014.

Optimization algorithm for data analysis

One of the methods to find this equivalent is to focus on the fact that (5. 0. 3) tasks have the following abilities.

It is registered as X K-αk ∇f (XK) $ + αk λ ψ (x), αK 2 (followed by a member αK ∇f (XK) 2 to connect Z). (5. 0. 3) The partial problem is the linear member ∇f (XK) T ( Z-XK) (a member close to the first Taylorsery brown), a member of the 2α1 K Z-XK 2 evaluation, the case of αK ↓ 0. , It became more difficult, and was made with members of the adjustment λ (X) of the exterior that did not change. In λ = 0, xk+1 = x k-αK ∇f (XK), and in this case, repetition (5. 0. 2) ((5. 0. 3)) is the steep slope descent method of section 4. Combined with a normal layout. The convergence of a subbilin speed (5. 0. 2) is indicated and the step length that has not been modified, which has a subbilin speed for a function F ((3. 3. 6)), which satisfies the nature of the lips of L. Select αk = 1/L. For confirmation, apply a "slope map" determined by 1 x-proxαλ ( x-α∇F (x)). (5. 0. 4) GA (x): Compared to = α (5. 0. 2), this map is performed on Itetion K as follows. I will determine the step: 1 k ( x-xk+1). (5. 0. 5) xk+1 = x k-αK Gαk (XK) The required properties of (x) are revealed. Remma 5

0 ∈ αλ∂ (Proxαλψ ( x-α∇F (x)) + (Proxαλψ ( x-α∇F (x)) - x-α∇F (x)). Definition (5. 0. 4) is applied. Then, ( x-α∇F (x)) = x-αGα (x) is 0∈αλ∂ ( x-αGα (x)) -α (x). ) ()

SJ 1-YJ (ATJ X-β),

By setting y = x - αGα (x) for any α∈(0, 1/l], we obtain Lα2 Gα (x) 2 2 (5. 0. 7) α F (x) - αGa (x) t ∇f (x) + ga (x) 2. x) The correct answer is: f (x - αGα (x)) f (x) - αGα (x) t ∇f (x) + ψ (z - αGα (x))

f (z) f (x) + ∇f (x) t (z - x), ψ (z) ψ (x - αGα (x)) + vt (z - (x - αGa (x)).

From part (a), we can see that v = (ga (x) - ∇f (x))/λ ∈ (x - αGα (x)). Following this choice v B (5. 0. 8) and using (5. 0. 7), we indeed get that for any α∈(0, 1/l], φ (x - αGa (x)) = f (x - αGa (x)) + λψ (x - αGα (x)) α F (x) - αGα (x) t ∇f (x) + Ga (x) 2 + λψ (x - αGa (x)) (from (5. 0. 7)) 2 α T F (x - αGa (x)) 2 + λψ (x - αGα (x)) (from (5. 0. 7)). (x) 2 2 + λψ (z) + (ga (x) - ∇f (x)) t (x - αGa (x) - z) (5. 0. 8)) α = f (z) + λψ (z) + ga (x) t (x - z) - ga ( X) 2, where the latter equality is in the footsteps of some members of the previous line. The best mean minimizer x∗ for the objective φ∗ (not without failures). Applying the proportionality (b) of this result, in fact the priority

This decreases, and the distance to the optimal x∗ also becomes smaller with every iteration. Substituting x=z=xk and α=αk into Lemma 5. 0. 6 and recalling (5. 0. 5), we get α φ (xk+1) = φ (xk - αk gaK (xk)) φ (xk) - k gαk (xk) 2, 2, which motivates the first application. In the 2nd adoption case, setting x = xk, α = αk, z = x∗ in Lemma 5. 0. 6 actually gives φ (xk) - φ∗. 0 φ (xk+1) - φ ＊ = φ (xk - αk gαK (xk)) - φ α gaK (xk) t (xk - x ∗) - k gαK (xk) 2 2 k ∗ 2 k = 2 k = 2 k = x - x - x ∗ - αk gaK (xk) 2 2 2aK 1 k = x - x ∗ 2 - xk+1 - x ∗ 2, 2aK<[XT a]l >Optimization algorithm for data analysis

follows xk+1 - x ∗ xk - x ∗. Substituting αK = 1/l into (5. 0. 10) and k = 0, 1, 2 ,... . . K - 1, the telescopic sum on the right hand side actually gives k-1.

L L 0 x - x∗2 - xk - x∗2 x0 - x∗2.

By the power of monotonicity

SJ 1-YJ (ATJ X-β),

This result follows immediately from the composition of the second-to-last equation.

6. In Section 4, the basic method of the rapid descent method to solve (1. 0. 1) for smooth F partial linino is a convex and linear, and when it is linear, positive γ. It was indicated that it would converge at the ratio of ( 1-γ // L) that satisfies (3. 3. 13) to L. This section shows that the use of the gradient information for more intelligents to achieve a higher convergence rate. The main idea is pulse. In the repetition of the pulse method, a small correction is added to the negative slope-focus estimated at the XK or the vicinity, and each repetition continues to move along the previous search direction. (Coole descent simply uses -∇f (XK) as a search direction). At first glance, it is not clear, but there is an intuitive background of the impulse idea. The step taken in the previous IC-1 is based on the information on the negative slope in this repetition and the previous repetition, that is, the search direction of the XK-2. Continuing the reasoning in the opposite direction indicates that the previous step is a linear combination of all gradient information encountered in all repetitions that start with the first repetition of the X0. If it is aggregated correctly, this information is just

XK + 1 = x k-αK ∇f (XK) + βK (X K-X K-1),

Stephen J. Wright < SPAN> In Section 4, the basic method of the rapid descent method to solve (1. 0. 1) to the smooth F partial linino is a convex and linear form. At that time, it was indicated that it converged ( 1-γ // L) percentage (3. 3. 13) to positive γ and L. This section shows that the use of the gradient information for more intelligents to achieve a higher convergence rate. The main idea is pulse. In the repetition of the pulse method, a small correction is added to the negative slope-focus estimated at the XK or the vicinity, and each repetition continues to move along the previous search direction. (Coole descent simply uses -∇f (XK) as a search direction). At first glance, it is not clear, but there is an intuitive background of the impulse idea. The step taken in the previous IC-1 is based on the information on the negative slope in this repetition and the previous repetition, that is, the search direction of the XK-2. Continuing the reasoning in the opposite direction indicates that the previous step is a linear combination of all gradient information encountered in all repetitions that start with the first repetition of the X0. If it is aggregated correctly, this information is just

XK + 1 = x k-αK ∇f (XK) + βK (X K-X K-1),

Stephen J. Light 6. Section 4, when the basic method of the rapid descent law to solve (1. 0. 1) for a smooth F partial linino is convex and is linear. It was indicated that the percentage of ( 1-γ // L) that satisfies (3. 3. 13) to positive γ and L is converged. This section shows that the use of the gradient information for more intelligents to achieve a higher convergence rate. The main idea is pulse. In the repetition of the pulse method, a small correction is added to the negative slope-focus estimated at the XK or the vicinity, and each repetition continues to move along the previous search direction. (Coole descent simply uses -∇f (XK) as a search direction). At first glance, it is not clear, but there is an intuitive background of the impulse idea. The step taken in the previous IC-1 is based on the information on the negative slope in this repetition and the previous repetition, that is, the search direction of the XK-2. Continuing the reasoning in the opposite direction indicates that the previous step is a linear combination of all gradient information encountered in all repetitions that start with the first repetition of the X0. If it is aggregated correctly, this information is just

XK + 1 = x k-αK ∇f (XK) + βK (X K-X K-1),

SJ 1-YJ (ATJ X-β),

Here, αK and βK are positive scalar. That is, βK (X K-X K-1) pulse lid members are added to the normal update of the descent. This method can be applied to any smooth convex focus F (which can also be applied to no n-flow functions), but the analysis of convergence is the easiest case in a highly convex secondary function ([38] See). This analysis also leads the correct value of the step length αK and βK). These properties are not unique to heavy ball methods, but are unique to other methods that use exercise. 6. 2. The c o-role gradient method is for solving linear systems for minimizing AX = B (or equally convex quadratic equation (6. 1. 2)), and a is symmetrical. It is positive defined. The conjunctive gradient is invented before the other algorithms featured in this section (see [27]), and the motivation is different, but it clearly uses the amount of exercise. The step format is (6. 2. 1)

Here PK = -∇F (XK) + Oring P K-1,

Regarding a value with an answer as αK, it matches (6. 1. 1) in the defined define of βK. The convex and strong secondary dysfunctional task (6. 1. 2) has excellent characteristics. This does not require preliminary knowledge of the range [γ, L].

Optimization algorithm for data analysis < Span> Here, αK and βK are positive scalar. That is, βK (X K-X K-1) pulse lid members are added to the normal update of the descent. This method can be applied to any smooth convex focus F (which can also be applied to no n-flow functions), but the analysis of convergence is the easiest case in a highly convex secondary function ([38] See). This analysis also leads the correct value of the step length αK and βK). These properties are not unique to heavy ball methods, but are unique to other methods that use exercise. 6. 2. The c o-role gradient method is for solving linear systems for minimizing AX = B (or equally convex quadratic equation (6. 1. 2)), and a is symmetrical. It is positive defined. The conjunctive gradient is invented before the other algorithms featured in this section (see [27]), and the motivation is different, but it clearly uses the amount of exercise. The step format is (6. 2. 1)

Here PK = -∇F (XK) + Oring P K-1,

Regarding a value with an answer as αK, it matches (6. 1. 1) in the defined define of βK. The convex and strong secondary dysfunctional task (6. 1. 2) has excellent characteristics. This does not require preliminary knowledge of the range [γ, L].

Optimization algorithm for data analysis Here, αK and βK are positive scalars. That is, βK (X K-X K-1) pulse lid members are added to the normal update of the descent. This method can be applied to any smooth convex focus F (which can also be applied to no n-flow functions), but the analysis of convergence is the easiest case in a highly convex secondary function ([38] See). This analysis also leads the correct value of the step length αK and βK). These properties are not unique to heavy ball methods, but are unique to other methods that use exercise. 6. 2. The c o-role gradient method is for solving linear systems for minimizing AX = B (or equally convex quadratic equation (6. 1. 2)), and a is symmetrical. It is positive defined. The conjunctive gradient is invented before the other algorithms featured in this section (see [27]), and the motivation is different, but it clearly uses the amount of exercise. The step format is (6. 2. 1)

Here PK = -∇F (XK) + Oring P K-1,

Regarding a value with an answer as αK, it matches (6. 1. 1) in the defined define of βK. The convex and strong secondary dysfunctional task (6. 1. 2) has excellent characteristics. This does not require preliminary knowledge of the range [γ, L].

Optimization algorithm for data analysis

The spectrum A of its own value selects the length of step AK and Oring adaptively. (In fact, αK is selected as a strict minimum according to the search direction PK). The main calculation in each repetition is the multiplication of a matrix and a vector with a participation A, which is equivalent to the cost of evaluating the gradient of the FB B B B B B B B B B B B B B BB (6. 1. 2). The most important thing is that the convergence theory is abundant, and the convergence is characterized by the characteristics of all species of A (as well as extreme factors). It is guaranteed that the convergence to the strict decision (6. 1. 2) is below NTES (of course, the operation is made accurately). In recent years, research has been actively conducted to extend the c o-role slope to general smooth function F. Such a "non-linear" method of the co-stammed slope is distinguished by the selection of ORING, which is more important, and more important. In the latter case, the spectrum A with a unique value in each direction of the search is adapted to the length of step AK and Oring. (In fact, αK is selected as a strict minimum according to the search direction PK). The main calculation in each repetition is the multiplication of a matrix and a vector with a participation A, which is equivalent to the cost of evaluating the gradient of the FB B B B B B B B B B B B B B BB (6. 1. 2). The most important thing is that the convergence theory is abundant, and the convergence is characterized by the characteristics of all species of A (as well as extreme factors). It is guaranteed that the convergence to the strict decision (6. 1. 2) is below NTES (of course, the operation is made accurately). In recent years, research has been actively conducted to extend the c o-role slope to general smooth function F. Such a "non-linear" method of the co-stammed slope is distinguished by the selection of ORING, which is more important, and more important. In the latter case, spectrum A, which has a unique R value in each direction of search, adaptively selects the length of step AK and Oring. (In fact, αK is selected as a strict minimum according to the search direction PK). The main calculation in each repetition is the multiplication of a matrix and a vector with a participation A, which is equivalent to the cost of evaluating the gradient of the FB B B B B B B B B B B B B B BB (6. 1. 2). The most important thing is that the convergence theory is abundant, and the convergence is characterized by the characteristics of all species of A (as well as extreme factors). It is guaranteed that the convergence to the strict decision (6. 1. 2) is below NTES (of course, the operation is made accurately). In recent years, research has been actively conducted to extend the c o-role slope to general smooth function F. Such a "non-linear" method with a co-slope is distinguished by the selection of Oring, the accuracy of (6. 2. 1), and more important. The latter is R in each direction of search

The only difference is that the act of extracting XK → XK + βK (X K-X K-1) is performed before the gradient ∇f B (6. 3. 1) is evaluated, during that time (6. The gradient of 1. 1) is primitive in point XK. For the analysis (and implementation), fix αk ≡ 1/l and rewrite the update (6. 3. 1) as follows: (6. 3. 3. 2a) (6. 3. 2B) )

1 xk+1 = yk - ∇f (yk), l yk+1 = xk+1+βk+1 (xk+1 - xk),

Stephen J.

SJ 1-YJ (ATJ X-β),

We have obtained the convergence of the NESTEROV scheme on convex focus aggregates. Confirm using the formulated [3] discussion in 7, Section 3. 7]. This test is technical, like LED, and is not flashy. Recently, the concrete progress has been seen to withdraw the similar algorithm (6. 3. 2), and they have plausible geometric and algebraic legitimacy; [8, 21]. Axiom 6. F (1. 0. 1) is actually convex, and ∇f does not change (like (3. 3. 6)), and the lips are continuously differentiated, and the minimum amount F is pointed. Assuming that it reaches with x ∗:. During this time, the method defined (6. 3. 2) and (6. 3. 3) defined as x0 = y0 is a repetitive priority order with the following characteristics: 2LX 0-x * 2, t = 1, 2. , ... (T + 1) 2 proof. From the uneven F and (3. 3. 7), for all X and y in the footprint, in fact F (XT) -F ∗.

f (y - ∇f (y)/L) -F (x) f (y -∇f (y)/l) - F (y) + ∇f (y) t ( y-x) l ∇f ( y) t (y - ∇f (y)/ l-y) + Y -∇f (y)/ l-y2 + ∇f (y) t ( y-x) 2 1 = - ∇f (y) 2 + ∇f (y) t (y). 2L k If this edge is y = y, x = xk (6. 3. 6).

f (xk + 1) -F (xk) = f (yk -∇f (yk)/L) -F (xk) 1 ∇f (yk) 2 + ∇f (y k-xk) 2L L L L L L = -xk+ 1-YK 2-l (xk+ 1-yk) t (y k-xk). 2 k ∗ Subasus (6. 3. 6) to (6. 3. 6), (6), (6), (6) 3. 2a) L (6. 3. 8) f (6. 3. 8) -F (x ∗) -xk + 1-Yk 2-l (xk+ 1-yk) t (y k-x ∗) ) Is obtained. 2 ΔK: = f (xk) -F (x *) is introduced, (6. 3. 7), λk+1-1-1 (6. 3. 8), and (6. 3. 7) Get).

(λk+1-1-1) (ΔK+ 1-ΔK)+ΔK+1 L-λK+1 xk+ 1-YK 2-L (xk+ 1-yk) T (λk+1 yk-- (λK+1--) 1-1-1 1) X K-X *) is obtained. 2

Optimization algorithm for data analysis

If you multiply (6. 3. 4) on this restriction and apply (6. 3. 4), you will get λ2k+1 ΔK+ 1-λ2k ΔK (6. 3. 9). 9) L-λk+1 (xk+ 1-yk) 2+2λK+1 (xk+ 1-yk) t (λk+1 yk-- (λk+1-1) x k-x ∗) 2 l = -λk+ 1 xk+1- (λk+1-1) x k-x ∗ 2-λk +1 yk-- (λK+1-1) X K-x ∗ 2, 2 Here, in the last equation, similar A2+2AT B = A + B 2-B2 is applied. Λk+2 is applied to (6. 3. 2b) and λk+2 βK+1 = λk+1-1-1-1-1-1-1-1 from (6. 3. 3), λK+2 yk+1 = λk+2 xk+ 1+λk+2 βK+1 (xk+ 1-XK) = λk+2 xk+1+(λk+1-1) (xk+ 1-xk) is obtained. If you sort this equation, it will be λk+1 xk+1- (λk+1-1) xk = λk+2 yk+1- (λk+2-1) xk+1. Substitute the 1st paragraph of the members on the right side (6. 3. 9) and apply the definition (6. 3. 10).

SJ 1-YJ (ATJ X-β),

L λ2k+1 ΔK+ 1-λ2k ΔK- (UK+1 2-UK 2). 2 If K = 0, 1,. Λ0 = 0 is applied to both sides of this inequality, L L λ2t ΔT (U0 2-UT 2) x 0-x ∗ 2, 2 2, 2, 2, 2, 2, 2 2, 2 2, LX 0-X ∗ ∗ ∗ ∗ ∗ It becomes 2. (6. 3. 11) Δt = f (XT) -F (x ∗) 2λ2t

The simple induction method effectively approves λK (k + 1)/2 for k = 1, 2, ... Then, the description of the axians follows the footprints that replaced this plane in (6. 3. 11). 6. Acceleration Nesteloff gradient: If you are strongly convex: Feel (3. 2. 5) to γ & amp; amp; gt; 0 (3. 2. 5) Let's continue calculating the Nes telev. This method actually uses the same update (6. 3. 2) as the case in the case of weak convexity, and is the same for initialization, but the choice of βK+1 is different: √ √ √. L-γ κ-1 (6. 4. 1) βk+1 ≡. Check the convergence result. Prix 6. 4. 2. In fact, ∇f is a constant L and continuously can be continuously differentiated according to lipsutz, and it is strongly convex with coarse coefficient γ, and uniquely.

Stephen J.

Maximizes. During this time, the method of holding the starting point x0 = y0 (6. 3. 2), (6. 4. 1), L+γ 0 1 T T ∗ ∗ 2 x 1 -√ f (x), f (x), Fill T = 1, 2, ... 2 κ confirmation. Confirmation uses the subtraction value of the industrial function φk (Z), which is determined as follows: γφ0 (z) = f (y0) + z-y0 2, √2 φk + 1 (z) = (1-1 / κ) φk (Z) 1 γ + √ f (yk) + ∇f (yk) t ( z-yk) + z-yk 2. 2k

Each φk (-) is a secondary function, and from a inductive discussion, it can be seen that ∇2 φk (Z) = γi for all K and all Z. Here, VK is the minimum φk (-), and φ * k is the optimal value. (6. 4. 3a), V0 = y0). Note that φk is approximated to F as it becomes K → ∞. To show this, replace the last member in parentheses (6. 4. 3b) in parentheses (6. 4. 3b) using (3. 3. 9), and from both sides (6. 4. 3b). ) √ (6. 4. 5) φk+1 (Z) -F (Z) -F (Z) (1-1/ κ) (φk (Z) -F (Z)). In the remaining part of the evidence, the following edges are executed: f (XK) - φk (z) = φ φ ∗ k,

What gives f (z) -F (x ∗) (l/2) z-x ∗ 2? F (XK) -F (x *) φ * k-f (x *) φ k (x *) φ k (x *) - f (x *) - f (XK) φ * k-f (x *), x *) √ (1-1/ κ) k (φ0 (x *) -F (x *)) (6. 4. 7) can be obtained. 4. 7) √ (1-1/ κ) k [(φ0 (x *) -F (x0)) + (f (x0) -F (x *)) √ γ + L 0 x-x * 2 (1-1/ κ) K 2 certificate ends with the establishment of (6. 4. 6) and is inducted down according to K. Since it is x0 = y0, K = 0 is fair due to the definition.

1 ∇f (YK) 2L √ √ √ √ 1 = (1-1/ κ) F (XK) + (XK) + (1-1/ κ) (F (YK) -F (XK)) + F (YK)/ κ -∇f (yk) 2L √ √ √ √ 1 (1-1/ κ) φ * k + (1-1/ κ) ∇f (yk) t (y k-xk) + f (yk)/ κ-κ -κ F (YK) 2. 2L F (YK)

Optimizatio n-algorithm for data analysis < SPAN> Each φk ( -) is a secondary function, and for all K and all Z, ∇2 φk (Z) = γi for all K and all Z. You can see that it will be. Here, VK is the minimum φk (-), and φ * k is the optimal value. (6. 4. 3a), V0 = y0). Note that φk is approximated to F as it becomes K → ∞. To show this, replace the last member in parentheses (6. 4. 3b) in parentheses (6. 4. 3b) using (3. 3. 9), and from both sides (6. 4. 3b). ) √ (6. 4. 5) φk+1 (Z) -F (Z) -F (Z) (1-1/ κ) (φk (Z) -F (Z)). In the remaining part of the evidence, the following edges are executed: f (XK) - φk (z) = φ φ ∗ k,

What gives f (z) -F (x ∗) (l/2) z-x ∗ 2? F (XK) -F (x *) φ * k-f (x *) φ k (x *) φ k (x *) - f (x *) - f (XK) φ * k-f (x *), x *) √ (1-1/ κ) k (φ0 (x *) -F (x *)) (6. 4. 7) can be obtained. 4. 7) √ (1-1/ κ) k [(φ0 (x *) -F (x0)) + (f (x0) -F (x *)) √ γ + L 0 x-x * 2 (1-1/ κ) K 2 certificate ends with the establishment of (6. 4. 6) and is inducted down according to K. Since it is x0 = y0, K = 0 is fair due to the definition.

1 ∇f (YK) 2L √ √ √ √ 1 = (1-1/ κ) F (XK) + (XK) + (1-1/ κ) (F (YK) -F (XK)) + F (YK)/ κ -∇f (yk) 2L √ √ √ √ 1 (1-1/ κ) φ * k + (1-1/ κ) ∇f (yk) t (y k-xk) + f (yk)/ κ-κ -κ F (YK) 2. 2L F (YK)

Optimizatio n-Each algorithm for data analysis is a secondary function, and from inductive discussions, it can be ∇2 φk (Z) = γi for all K and all Z. I understand. Here, VK is the minimum φk (-), and φ * k is the optimal value. (6. 4. 3a), V0 = y0). Note that φk is approximated to F as it becomes K → ∞. To show this, replace the last member in parentheses (6. 4. 3b) in parentheses (6. 4. 3b) using (3. 3. 9), and from both sides (6. 4. 3b). ) √ (6. 4. 5) φk+1 (Z) -F (Z) -F (Z) (1-1/ κ) (φk (Z) -F (Z)). In the remaining part of the evidence, the following edges are executed: f (XK) - φk (z) = φ φ ∗ k,

What gives f (z) -F (x ∗) (l/2) z-x ∗ 2? F (XK) -F (x *) φ * k-f (x *) φ k (x *) φ k (x *) - f (x *) - f (XK) φ * k-f (x *), x *) √ (1-1/ κ) k (φ0 (x *) -F (x *)) (6. 4. 7) can be obtained. 4. 7) √ (1-1/ κ) k [(φ0 (x *) -F (x0)) + (f (x0) -F (x *)) √ γ + L 0 x-x * 2 (1-1/ κ) K 2 certificate ends with the establishment of (6. 4. 6) and is inducted down according to K. Since it is x0 = y0, K = 0 is fair due to the definition.

SJ 1-YJ (ATJ X-β),

Optimizatio n-algorithm for data analysis

Therefore, if the right side B (6. 4. 8) is on top of φ * k+1, the approval is established (the theorem is proven). Remembering the observation (6. 4. 4), the conductive function of both sides (6. 4. 3b) by Z is √ √ √ (6. 4. 9) ∇k+1 (Z) = γ (1-1 1 /is κ) (z) (z) -VK) + ∇f (yk)/κ + γ ( z-yk)/κ. VK+1 is the minimum price of φk+1, so if ∇k+1 (VK+1) = 0 is installed in (6. 4. 9), √ (6. 4. 10) vk+1 = (1-1 / κ) vk + yk / κ -∇f (yk) / (γ κ) is obtained. If you draw yk from both sides of this equation and tak e-2 from both sides, √ vk + 1-yk 2 = (1-1 / κ) 2 y k-vk 2 + ∇f (yk) 2 / (γ2 κ) (γ2 κ) (γ2 κ) 6. 4. 11) √ -2 (1-1/ κ)/ (γ κ) ∇f (yk) t (v k-yk). Evaluation of φk+1 in z = yk using (6. 4. 4) and (6. 4. 3b), γ γ φ+y k-vk+1 2 2 2 2 √ = (1-11 / κ) φk (yk (yk)) + f (yk)/ κ (6. 4. 12) is obtained. 4. 12) √ γ γ = (1-1/ κ) φ φ k + (1-1/ κ) Y K-vk 2 + f (YK)/ κ. 2 (6. 4. 11) (6. 4. 12) When assigned to √ √ √ k + 1 = (1-1/ κ) φ * k + f (YK)/ κ + γ (1-1/ κ)/ (2 κ) y k-vk 2 √ 1 - ∇ F (YK) 2 + (1-1/ κ) ∇f (yk) t (v k-yk)/ κ 2。 (6. 4. 13) is obtained. 4. 13) √ (1-1/ κ) φ φ ∗ k + f (YK)/ κ √ 1 -∇f (yk) 2 + (1-1/ κ) ∇f (yk) t (v k-yk) / κ, 2L Here, I just thrown a no n-negative member on the right side to get an inequality. The rest is √ (6. 4. 14) v k-yk = κ (Y K-XK). v0 = x0 = y0, so

Stephen J.

In fact, φ * k + 1 is regarded as the upper world of the right side (6. 4. 8). As a result, (6. 4. 6) is justified, and the confirmation of the axians is completed. 6. 5. The lower world regarding the "optimal" speed of the right method of NESTEROV is the best (up to the constant accurate to constant (constant quotes) among algorithms that use gradient information in XK times. It is used for the fact that it is considered to be. This statement can be proven by the support of the finely developed functions, and on the other hand, how to use all slope (∇f (XI), I = I = I = I =. Use only 0, 1, 2, ..., K. K.). ), It has the ability to gain priority over 6. 3. 5. The functions proposed in 32] were convex Fi (x) = (1/2) XT A X-ET1 X, and here ⎡ 2-1 0. 0 ⎥ ⎥ 1 ⎢ 1 ⎢ 2-1 0.. ... 0 ⎥ ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥, e1 = ⎥ ⎢ .. .. ⎥ ⎢. ⎥ ⎢ ⎢ ⎢ .. ⎥ ⎥ ⎥ ↪ ↪ ⎥ ⎥ ↪ sm_23a3 ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ ⎣ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 結 1- I/(n+1), I = 1, 2, ............................................... J ξ J ξ Same to 0.

In the case of J = 0, 1, ..... K for a certain coefficient CIL, each XK ITEITE may only have an unmimulated record in the first K components. You can see that there is. Dropping is in the footprints, and in fact, in such a method, 2 n j 1- (x ∗ j) 2 =. (6. 5. 1) x k-x ∗ 2 j = k+1

A minor mathematics indicates that (6. 5. 2) is actually (6. 5. 2).

Can be indicated.

3L x 0-x * 2, 32 (k + 1) 2

SJ 1-YJ (ATJ X-β),

Optimization-algorithm for data analysis

7. Newton's methods are methods that use first-order information (gradients or subgradients) on the target function. We have shown that such algorithms can give sequences of iterations that converge at linear or subcritical speed. In this chapter, we turn to methods that use information on second derivatives (Guessians). The archetypal method here is Newton's method, named after Isaac Newton, who proposed a method for polynomial equations around 1670. For many functions, including those that arise in analyzing data, it is easy to compute second-order information, in the sense that the functions we deal with are simple (usually compositions of elementary functions). When comparing first-order methods, a compromise must be made. Second-order methods usually have a local superlinear or quadratic retirement rate: they converge faster once the iterations reach an environment of solutions that satisfy sufficient quadratic conditions. Moreover, their property of global convergence is attractive: with appropriate refinements, it can be proven that they do not converge to a saddle point. However, they come at a high cost of computing and processing the second-order information, as well as the step calculations. Consider the problem of subtracting f(x), where this compromise is:

Here, F: RN → R is twice as much as the lips of the lips, and Gessian contains the lon g-term lips (7. 1. 2).

∇2 f (x) - ∇2 f (x) m x-x,

Her e-is the vector's Euclidnorm and its induction matrix Norm. The Newton Law is K = 0. 1. 2.

Stephen J.

The approximation of the number TAYLOR 2 is an approximation F of the current XK ITRIT Y-CONTRIBUTIONS of 1 F (XK + P) ≈ F (XK) + ∇f (XK) t p + pt ∇2 f (xk) p. If 2 ∇2 f (XK) is exactly determined, the PK on the right is the only one.

PK = -∇2 f (XK) -1 ∇f (XK).

This is the Newton's step. In this way, the method of Newton is guided by the correct repetition: XK+1 = XK -∇2 f (XK) -1 ∇f (XK).

We have obtained local convergence in the District of Point X and meet the required conditions 2 or more. Let's consider the issues of Prior 7. (7. 1. 1). In this task, F is separated twice by LipsHitsa without changing the lips of M. During this time, x 0-x ∗ 2m ∗ (7. 1. 5) converges to X with a secondary curve, and is mk x-x ∗ 2, k = 0, 1, 2, ... ... ... (7. 1. 7) XK+ 1-x γ γ-confirmation. If ∇f (x *) = 0 is used from (7. 1. 4) and (7. 1. 5), xk+1 --x * = x k-x * - ∇2 f (XK) -1 ∇f (XK) = ∇2 f (xk) -1 [∇2 f (XK) (x k-x *) - (∇f (XK) -∇f (x *)]. For example, (7. 1. 8)

XK+1 --x ∇2 F (XK) -1 ∇2 f (XK) (X K-X *) - (∇f (XK) -∇f (X *)).

1 K ∗ ∇f (x) - ∇f (x) = ∇2 f (XK + + + + + (xk), 1 k ∗ ∇f (x) = ∇f (x) = ∇f (x), by using Taylor Areas (XK, P = X ∗ -XK). T (x ∗ -xk)) (x k-x ∗) DT appears. 0

Using this result and the lip conditions (7. 1. 2), the following are as follows.

∇2 f (xk) (x k-x ∗) - (∇f (xk) - ∇f (x ∗)) $ $ 1 $ = $ [∇2 f (xk) - ∇2 f (xk + t (xk) ∗ -xk)] (x k-x ∗) dt $ 0 $ 0 1 ∇2 f (xk) - ∇2 f (xk + t (xk) x k-x ∗ dt 0

Mt dt X K-x ∗ 2 = 12 mx k-x ∗ 2.

Since weIland t-GoffMann is inequality [28] and (7. 1. 2), it is actually ｜min (∇2 f (XK)) - λmin (∇2 f (x ∗)) | ∇ 2 F (xk) - ∇2 f (x ∗) mx k-x ∗,

SJ 1-YJ (ATJ X-β),

Here, λmin (-) indicates the meaningless meaning of the symmetric queue. In this way, γ, λmin (∇2 F (XK)) λmin (∇2 f (x *)) can be obtained for γ (7. 1. 1. 10) x k-x *, 2m. -MX K-x ∗ γ γ-m 2m 2m 2m 2m 2m 2m 2m, so it is substantially ∇2 f (XK) -1 2/γ. If this result (7. 1. 9) is assigned to (7. 1. 9), it becomes 2m k mk x x x ∗ 2 = x-x ∗ 2, xk+ 1-x γ 2 γ, which is actually local secondary. The speed is approved for convergence. If you use (7. 1. 10) again, it will be mk 1 x-x k-x k-x ∗, xk+ 1-x γ 2, and as a result, you will actually infer (7. As long as 1. 10) is satisfied, it is approved that the set will converge to x ∗. Of course, it is not necessary to clearly define the starting point X0 in the specified convergence area. The priorities approaching X * will eventually enter this area, and during that time, a warranty of secondary convergence works. In fact, we prove that the Newton Law is only repeated in the area of point x * and satisfies the necessary optimal conditions 2, so that it will converge quickly. But what happens when you start far from a similar point? 7. 2. If the Newton Law Function F to the convex function is not only convex but also smooth, it is possible to devise a Newton method option, and in addition to local secondary descent, mass convergence and difficulty. Such results can be justified (especially results based on verse 4. 5). If f is convex with γ-module, and if it satisfies the continuity of the lips (3. 3. 6), Gessian ∇2 f (XK) is justified to all K and Wi.

Stephen J. Wright < SPAN> min (-) here shows the meaningless meaning of the symmetric procession. In this way, γ, λmin (∇2 F (XK)) λmin (∇2 f (x *)) can be obtained for γ (7. 1. 1. 10) x k-x *, 2m. -MX K-x ∗ γ γ-m 2m 2m 2m 2m 2m 2m 2m, so it is substantially ∇2 f (XK) -1 2/γ. If this result (7. 1. 9) is assigned to (7. 1. 9), it becomes 2m k mk x x x ∗ 2 = x-x ∗ 2, xk+ 1-x γ 2 γ, which is actually local secondary. The speed is approved for convergence. If you use (7. 1. 10) again, it will be mk 1 x-x k-x k-x ∗, xk+ 1-x γ 2, and as a result, you will actually infer (7. As long as 1. 10) is satisfied, it is approved that the set will converge to x ∗. Of course, it is not necessary to clearly define the starting point X0 in the specified convergence area. The priorities approaching X * will eventually enter this area, and during that time, a warranty of secondary convergence works. In fact, we prove that the Newton Law is only repeated in the area of point x * and satisfies the necessary optimal conditions 2, so that it will converge quickly. But what happens when you start far from a similar point? 7. 2. If the Newton Law Function F to the convex function is not only convex but also smooth, it is possible to devise a Newton method option, and in addition to local secondary descent, mass convergence and difficulty. Such results can be justified (especially results based on verse 4. 5). If f is convex with γ-module, and if it satisfies the continuity of the lips (3. 3. 6), Gessian ∇2 f (XK) is justified to all K and Wi.

Stephen J. Wright Here λmin (-) indicates the meaningless meaning of the symmetric procession. In this way, γ, λmin (∇2 F (XK)) λmin (∇2 f (x *)) can be obtained for γ (7. 1. 1. 10) x k-x *, 2m. -MX K-x ∗ γ γ-m 2m 2m 2m 2m 2m 2m 2m, so it is substantially ∇2 f (XK) -1 2/γ. If this result (7. 1. 9) is assigned to (7. 1. 9), it becomes 2m k mk x x x ∗ 2 = x-x ∗ 2, xk+ 1-x γ 2 γ, which is actually local secondary. The speed is approved for convergence. If you use (7. 1. 10) again, it will be mk 1 x-x k-x k-x ∗, xk+ 1-x γ 2, and as a result, you will actually infer (7. As long as 1. 10) is satisfied, it is approved that the set will converge to x ∗. Of course, it is not necessary to clearly define the starting point X0 in the specified convergence area. The priorities approaching X * will eventually enter this area, and during that time, a warranty of secondary convergence works. In fact, we prove that the Newton Law is only repeated in the area of point x * and satisfies the necessary optimal conditions 2, so that it will converge quickly. But what happens when you start far from a similar point? 7. 2. If the Newton Law Function F to the convex function is not only convex but also smooth, it is possible to devise a Newton method option, and in addition to local secondary descent, mass convergence and difficulty. Such results can be justified (especially results based on verse 4. 5). If f is convex with γ-module, and if it satisfies the continuity of the lips (3. 3. 6), Gessian ∇2 f (XK) is justified to all K and Wi.

Stephen J.

If the line branch length αk = 1 v (4. 0. 1) is corrected by the linear search framework that meets the weak conditions of Wulf (4. 5. 2), the local secondary convergence characteristics of 7. 1. 6 are the same. The convergence of the mass is enhanced. (A discussion based on Taylor's worth (theorem 3. 3. 1) is r e-indicated that these conditions αk = 1 are filled with all XK closest to the minimum minimum X ∗). Here, let's take a look when F meets conditions (3. 3. 6) in Confaval, but not considered a tight convex. Here, Gessian ∇2 f (XK) may be unique to a certain K, and for this purpose (7. 1. 4), it may not be absolutely uniquely. However, if you add an arbitrary positive amount λk & amp; gt; 0 to the diagonal, it will actually be determined by (7. 2. 1), which is actually a modified goal of Newton. You can.

PK = -[∇2 F (XK) + λk I] -1 ∇f (XK), < SPAN> Length αK = 1 V (4. 0. 1) are weak conditions for Wulf (4. 5. 2) If you modify the framework of satisfying linear search, the convergence of these masses will be enhanced by the local secondary convergence characteristics of the axiac 7. 1. 6. (A discussion based on the axians of the Taylor (theorem 3. 3. 1) is r e-indicated that these conditions αk = 1 are satisfied with all XKs closest to the minimum minimum X ∗). Here, let's take a look when F meets conditions (3. 3. 6) in Confaval, but not considered a tight convex. Here, Gessian ∇2 f (XK) may be unique to a certain K, and for this purpose (7. 1. 4), it may not be absolutely uniquely. However, if you add an arbitrary positive amount λk & amp; gt; 0 to the diagonal, it will actually be determined by (7. 2. 1), which is actually a modified goal of Newton. You can.

PK = -[∇2 F (XK) + λk I] -1 ∇f (XK), li n-length αk = 1 v (4. 0. 1) satisfy the weak conditions of Wulf (4. 5. 2) If the framework of linear search is modified, the local secondary convergence characteristics of the axiotics 7. 1. 6 will enhance the convergence of these masses. (A discussion based on Taylor's worth (theorem 3. 3. 1) is r e-indicated that these conditions αk = 1 are filled with all XK closest to the minimum minimum X ∗). Here, let's take a look when F meets conditions (3. 3. 6) in Confaval, but not considered a tight convex. Here, Gessian ∇2 f (XK) may be unique to a certain K, and for this purpose (7. 1. 4), it may not be absolutely uniquely. However, if you add an arbitrary positive amount λk & amp; gt; 0 to the diagonal, it will actually be determined by (7. 2. 1), which is actually a modified goal of Newton. You can.

PK = -[∇2 F (XK) + λk I] -1 ∇f (XK),

In any η (0, 1) b (4. 5. 1), if λk is enough to λk /(l + λk) η, the conditions of (4. 5. 1) are satisfied. Because it is possible, you can get the obtained direction PK in the framework of linear search in verse 4. 5, and if there is an X * of (1. 0. 1), you can obtain a method of converging it to solve it at any time. Furthermore, if the minimum value x * is the only one, and if ∇2 f (x *) is exactly determined by the secondary sufficient conditions, ∇2 f (XK) is quite a lot. It is exactly determined when it is large. Therefore, if η is sufficient, the condition (λk = 0) is satisfied ((7. 2. 1)) (λk = 0) in Newton's uncertain direction ((7. 2. 1)). (7. 2. 1) is used in the framework of linear search in verse 4. 5, λk = 0, and if possible (4. 5. 2) the step length when satisfying (4. 5. 2) is αK = 1, and is promised by the aorium 4. 5. In addition to the larg e-range convergence and complexity, local secondary convergence to X *can be obtained. 3. 7. 3. Newton method smooth and no n-vulnerable functions for no n-vulnerable functions F Gessian ∇2 f (XK) may be uncertain for K. The direction of Newton (7. 1. 4) does not exist (when ∇2 f (XK) is unique) or not (when ∇2 f (XK) has a negative value)). However, the modified direction of Newton can be used.

Optimization algorithm for data analysis < Span> In any η (0, 1) b (4. 5. 1), if λk is large enough for λk /(l + λk) η, Since the conditions of (4. 5. 1) are also satisfied, put the obtained PK in the framework of linear search in verse 4. 5 and solve it at any time if (1. 0. 1) exists. You can get a way to converge. Furthermore, if the minimum value x * is the only one, and if ∇2 f (x *) is exactly determined by the secondary sufficient conditions, ∇2 f (XK) is quite a lot. It is exactly determined when it is large. Therefore, if η is sufficient, the condition (λk = 0) is satisfied ((7. 2. 1)) (λk = 0) in Newton's uncertain direction ((7. 2. 1)). (7. 2. 1) is used in the framework of linear search in verse 4. 5, λk = 0, and if possible (4. 5. 2) the step length when satisfying (4. 5. 2) is αK = 1, and is promised by the aorium 4. 5. In addition to the larg e-range convergence and complexity, local secondary convergence to X *can be obtained. 3. 7. 3. Newton method smooth and no n-vulnerable functions for no n-vulnerable functions F Gessian ∇2 f (XK) may be uncertain for K. The direction of Newton (7. 1. 4) does not exist (when ∇2 f (XK) is unique) or not (when ∇2 f (XK) has a negative value)). However, the modified direction of Newton can be used.

Optimization algorithm for data analysis In any η (0, 1) b (4. 5. 1), the λk is enough for λk /(l + λk) η, (4. 5. 1) Conditions are also satisfied, so put the obtained PK in the framework of linear search in verse 4. 5, and converge to solve it at any time if X * (1. 0. 1) exists. You can get a method. Furthermore, if the minimum value x * is the only one, and if ∇2 f (x *) is exactly determined by the secondary sufficient conditions, ∇2 f (XK) is quite a lot. It is exactly determined when it is large. Therefore, if η is sufficient, the condition (λk = 0) is satisfied ((7. 2. 1)) (λk = 0) in Newton's uncertain direction ((7. 2. 1)). (7. 2. 1) is used in the framework of linear search in verse 4. 5, λk = 0, and if possible (4. 5. 2) the step length when satisfying (4. 5. 2) is αK = 1, and is promised by the aorium 4. 5. In addition to the larg e-range convergence and complexity, local secondary convergence to X *can be obtained. 3. 7. 3. Newton method smooth and no n-vulnerable functions for no n-vulnerable functions F Gessian ∇2 f (XK) may be uncertain for K. The direction of Newton (7. 1. 4) does not exist (when ∇2 f (XK) is unique) or not (when ∇2 f (XK) has a negative value)). However, the modified direction of Newton can be used.

Optimization algorithm for data analysis

Equation (7. 2. 1) -There is no way to convert the direction of Newton to ensure the descent in the linear search frame. Other layouts are described in [36, Leader 3]. Because they only need to continue facts (that is, to avoid the extraction of square root from negative numbers), add positive ingredients to diagonal and numbers in the factory. It is necessary to convert the decomposition, and when calculating Newton's step PK, it returns ∇2 f (XK) after applying the modified factor decomposition. Another method is ∇2 f (XK) = QK λk Qtk (here, QK is a orthogonal matrix, λK is a corrected version of λK, and there are personal values in it). Calculating disassembly on personal values and recognizing all diagonal λ positively. During this time, in accordance with (7. 1. 4), PK can be modified as ~-1 Qtk ∇f (XK). PK: = -qk λ K ~ K, we are deemed to meet some η & amp; amp; gt; 6-point class xˆ, that is, ∇f (ˆx) = 0. The more powerful guarantee can be obtained from the confidential areas version of the Newton Law, which guarantees the convergence of nearly two important conditions, that is, to satisfy ∇2 f (ˆx) = 0. < SPAN> equation (7. 2. 1) -There is no way to convert the direction of Newton to ensure the descent in the linear search. Other layouts are described in [36, Leader 3]. Because they only need to continue facts (that is, to avoid the extraction of square root from negative numbers), add positive ingredients to diagonal and numbers in the factory. It is necessary to convert the decomposition, and when calculating Newton's step PK, it returns ∇2 f (XK) after applying the modified factor decomposition. Another method is ∇2 f (XK) = QK λk Qtk (here, QK is a orthogonal matrix, λK is a corrected version of λK, and there are personal values in it). Calculating disassembly on personal values and recognizing all diagonal λ positively. During this time, in accordance with (7. 1. 4), PK can be modified as ~-1 Qtk ∇f (XK). PK: = -qk λ K ~ K, we are deemed to meet some η & amp; amp; gt; 6-point class xˆ, that is, ∇f (ˆx) = 0. The more powerful guarantee can be obtained from the confidential areas version of the Newton Law, which guarantees the convergence of nearly two important conditions, that is, to satisfy ∇2 f (ˆx) = 0. Equation (7. 2. 1) -There is no way to convert the direction of Newton to ensure the descent in the linear search frame. Other layouts are described in [36, Leader 3]. Because they only need to continue facts (that is, to avoid the extraction of square root from negative numbers), add positive ingredients to diagonal and numbers in the factory. It is necessary to convert the decomposition, and when calculating Newton's step PK, it returns ∇2 f (XK) after applying the modified factor decomposition. Another method is ∇2 f (XK) = QK λk Qtk (here, QK is a orthogonal matrix, λK is a corrected version of λK, and there are personal values in it). Calculating disassembly on personal values and recognizing all diagonal λ positively. During this time, in accordance with (7. 1. 4), PK can be modified as ~-1 Qtk ∇f (XK). PK: = -qk λ K ~ K, we are deemed to meet some η & amp; amp; gt; 6-point class xˆ, that is, ∇f (ˆx) = 0. The more powerful guarantee can be obtained from the confidential areas version of the Newton Law, which guarantees the convergence of nearly two important conditions, that is, to satisfy ∇2 f (ˆx) = 0.

1) satisfies the following equation (7. 3. 2).

[∇2 F (XK) + λi] DK = -∇F (XK), ∇2 f (XK) + λi

Here, λ is selected like this, and is actually considered a positive hal f-infinity, is 0 λ & amp; amp; gt; only in this case, k d = ΔK (see [31]). In this way, conclusions (7. 3. 1) were combined to explore the preferred meaning of scalar λk, and a special method for that was developed. In the case of larg e-scale problems, inference (7. 3. 1) has multiple factor decomposition of N × N matrix (and 7. 3. 2) to all possible values of λ). Because it is included, it may be almost very expensive. Known classification

SJ 1-YJ (ATJ X-β),

[∇2 F (XK) + λi] DK = -∇F (XK), ∇2 f (XK) + λi

Here, λ is selected like this, and is actually considered a positive hal f-infinity, is 0 λ & amp; amp; gt; only in this case, k d = ΔK (see [31]). In this way, conclusions (7. 3. 1) were combined to explore the preferred meaning of scalar λk, and a special method for that was developed. In the case of a larg e-scale problem, the reason (7. 3. 1) has multiple factor decomposition of the N × N matrix (and the coefficients of the coefficient of λ). Because it is included, it may be almost very expensive. Known classification

Stephen J. Light 1) meets the following equation (7. 3. 2).

[∇2 F (XK) + λi] DK = -∇F (XK), ∇2 f (XK) + λi

Here, λ is selected like this, and is actually considered a positive hal f-infinity, is 0 λ & amp; amp; gt; only in this case, k d = ΔK (see [31]). In this way, conclusions (7. 3. 1) were combined to explore the preferred meaning of scalar λk, and a special method for that was developed. In the case of larg e-scale problems, inference (7. 3. 1) has multiple factor decomposition of N × N matrix (and 7. 3. 2) to all possible values of λ). Because it is included, it may be almost very expensive. Known classification

Stephen J.

The "Dog Leg" method is considered to find a narrow inference (7. 3. 1) that can be applied when ∇2 f (XK) is defined. In this method, the curved paths covered by the conclusions of the λ-value in the section [0, ∞] (7. 3. 2) are higher than the normal method consisting of two segments. The first segment combines 0 and K point DK C, minimizes (7. 3. 1) along with the direction-f (x), and the second segment is DK C and Neutton's basis steps, Specifically, (7. 1. 4) is combined. In the proximity, the bathta that the "dog leg" path crosses the trust line D ΔK is accepted. If the Dogleg-if the line is completely off the inside of the confidential area, the DK is considered a Newton failure step. See 36, 4. 1. After explaining the subtitles (7. 3. 1) with a trusted area, we will explain how this subtitle can be applied as a perfect basis. The main roles are only the value of the reduction amount F predicted by (7. 3. 1) and the actual reduction amount F, and only F (XK) -F (XK + DK). It is. According to the standard, this correspondence should be close to 1. If it is a smaller tolerance error (for example, 10-4), we take a step and execute an appropriate repetition. In an unpleasant case, the radius of the trust area AK is very large, and to find a narrow inference (7. 3. 1) that can be applied when A < Span> ∇2 f (XK) is defined. Dog legs are considered. In this method, the curved paths covered by the conclusions of the λ-value in the section [0, ∞] (7. 3. 2) are higher than the normal method consisting of two segments. The first segment combines 0 and K point DK C, minimizes (7. 3. 1) along with the direction-f (x), and the second segment is DK C and Neutton's basis steps, Specifically, (7. 1. 4) is combined. In the proximity, the bathta that the "dog leg" path crosses the trust line D ΔK is accepted. If the Dogleg-if the line is completely off the inside of the confidential area, the DK is considered a Newton failure step. See 36, 4. 1. After explaining the subtitles (7. 3. 1) with a trusted area, we will explain how this subtitle can be applied as a perfect basis. The main roles are only the value of the reduction amount F predicted by (7. 3. 1) and the actual reduction amount F, and only F (XK) -F (XK + DK). It is. According to the standard, this correspondence should be close to 1. If it is a smaller tolerance error (for example, 10-4), we take a step and execute an appropriate repetition. In an unpleasant case, the radius of the trust area AK is very large, and to find a narrow inference (7. 3. 1) that can be applied when A∇2 f (XK) is defined, "dog legs". The law is considered. In this method, the curved paths covered by the conclusions of the λ-value in the section [0, ∞] (7. 3. 2) are higher than the normal method consisting of two segments. The first segment combines 0 and K point DK C, minimizes (7. 3. 1) along with the direction-f (x), and the second segment is DK C and Neutton's basis steps, Specifically, (7. 1. 4) is combined. In the proximity, the bathta that the "dog leg" path crosses the trust line D ΔK is accepted. If the Dogleg-if the line is completely off the inside of the confidential area, the DK is considered a Newton failure step. See 36, 4. 1. After explaining the subtitles (7. 3. 1) with a trusted area, we will explain how this subtitle can be applied as a perfect basis. The main roles are only the value of the reduction amount F predicted by (7. 3. 1) and the actual reduction amount F, and only F (XK) -F (XK + DK). It is. According to the standard, this correspondence should be close to 1. If it is a smaller tolerance error (for example, 10-4), we take a step and execute an appropriate repetition. In an unpleasant case, the radius of the trust area AK is very large, A

Then, decide how to proceed in this direction. The reliable method is the opposite, first select a distance AK, and then find a direction in which the length of this step can get the best progress. 7. 4. There are important advantages in the thir d-party supervisor approach of Newton's Trust District Law. A close approach based on the tertiary Supervisor has a similar nature, and there are some more complex guarantees. The Cuban regulator demands that Gessian is continuous in the lips, as in (7. 1. 2). It can be seen that the following tertiary functions gave the large upper world of F: 1 m (7. 4. 1) tm (z; x): = f (x) + ∇f (x) t (z- X) + ( Z-X) T ∇2 F (X) ( Z-X) + Z-x3. 2 6

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In particular, there is F (Z) TM (Z; X) for any X,

The basic tertiary adjustment algorithm starting from the X0 operates as follows: (7. 4. 2)

XK+1 = ARG Minus TM (Z; XK), Z

The characteristics of the complexity of this method are analyzed in [35], and the options are studied in [26] and [12, 13]. Instead of expressing the theory of techniques based on (7. 4. 2), a rudimentary algorithm is described using (7. 4. 1) and the theory of cool descent in verse 4. 1. Our algorithms are to find the necessary secondary conditions, that is, (7. 4. 3), almost satisfied.

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f (x + t1 u) - f (x) t (x + t2 (t1 /t2) u) - f (x) = 2 t1 t2 t f ((1 - t1 /t2) x + (t1 /t2) ( x + t2 u)) - f (x) = 2 t1 t2 (1 - t1 /t2) f (x) + (t1 /t2) f (x + t2 u) - f (x) t2 f (x + t2 u) - f (x) =. T2 In particular, 0 f (x + t (y - x)) for a little t & amp; amp; gt; 0, we seek, actually for all t & amp; amp; gt; 0 , we have f (x + t (y - x)). - Of the results presented here, most of our assertions are applicable in the overall hydraulic field (complete internal works) and do not require anything special in the final dimensional field, For simplicity, we use RN as the base field on which all functions and mass assignments are performed. 2. Properties of Convex Sets Convex sums have very favorable properties and can be effectively and elegantly outlined. It also provides some favorable properties with respect to its edges, and friends of friends. To this end, in this section we present basic rules for isolating and supporting hyperplanes of convex sets. The results begin with the confirmation that for any large convex quantity C there exists a unique (Euclidean) projection, and this precedent is supported by the fact that Basta The Hilbert or Banach field communities used for this purpose are sometimes needed to show that it is possible to separate them from the The discussions in these places and their people would, in fact, in principle, celebrate the affirmation that there is an all-emerging possibility for places of unlimited dimensions with rather ordered ways. The epigraph and gradient of the Introduction to Stochastic Optimization Lecture find many applications in the development of optimization algorithms and optimality proofs. A simple property of the large capacity of a convex function is that it is possible to First, if Ca is convex and large for each α∈A (A is an arbitrary index with large capacity), then there is a crossing point (Ca C = α∈A

This is also considered convex. Apart from this, the large amount of convex is closed at double scalar: if C is convex with α∈R, it is definitely a convex αc: = convex. The convex amount of mingkovsky 2 is represented by C1 + C2: =, which is also considered to be convex. To believe this, in the case of Xi, yi∈ci, λ (x1 + x2) + ( 1-λ) (y1 + y2) = λx1 + ( 1-λ) y1 + λx2 + ( 1-λ) y2∈ C1 + C2. ∈C1

In particular, the large capacity of the convex is closed by all linear synthesis: α∈rm, C = M I = 1 α α α-i CI is also considered convex. We are also a number of points X1,. XM ∈ RN is convex by m m Ov = λi xi: λi 0, λi = 1. I = 1

This large number is obviously convex. The projecting continues to consider the orthogonal projection of the large number of convex. Using this projection, you can pass the nature of the split and other convex data. See Figure 2. 2. 1.

Fig. 2. 2. 1. Dotting of point X to large cucimal C (with shelled πc (x))

Start with the traditional result of projecting zero landmarks into large numbers of convex. Axiki 2. 2. 2 (zero projection). C-Make a large number of closed balls without the starting point of the coordinates. 0 During this time, the bathta XC∈C is actually only XC 2 = INFX∈C X2. More than that, XC 2 = INFX 82 is only during this time when XC, Y-XC 0.

(2. 2. 3) About all y∈Cs.

Proof. Vice x, y, 1 1 x-y22 + x + y22 = x22 + y22. (Xn) ⊂c- is set as a set of c points, and XN 2 → m is set by n → φ. According to an arbitrary fou r-sided format (2. 2. 4) of any N and M∈N, we have 1 1 1 1 X N-XM 22 = XN 22 + XM 2 2-XN + XM 22. 2 2 Fixed & amp; gt; 0, we select n∈N like this, and actually enter the footprints of N, actually XN 22 M2 +. During this time, for each M, N N, we will use the large amount C unevenness of 1 1 1 1 1 1 X N-XM 22 2m2 + 2 - xn + XM 22. (2. 2. 5) 2 2

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Introduction to probability optimization

If XC 22 and T2 Y-XC 22 are drawn from both sides of the above inequality, -T2 Y-XC 22 22 2T XC, Y-XC. If you divide both sides of the inequality by 2T, it will be T-Y -XC 22 XC, Y-XC 2 for all t∈ (0, 1]. You can get it.

If this theorem is given, a simple transition will provide a more general demanding characteristics to the convex gathering. KOROLLARIA 2. 2. 6 (imaging to convex gathering). At this time, there is a single point πc (x) called X-πC (x) 2 = INFY EDC X-Y2, that is, πc (x) = Argminy Edc Y-x22. This projection is characterized by inequality πc (x) -x, y-πc (x) 0.

(2. 2. 7) About all y∈Cs. < SPAN> If the abov e-mentioned inequality (2. 2. 5) is used, 1 X N-XM 22 2m2 + 2-2m2 = 2. 2 √ Especially X N-XM 2 2; XN was optional. (XN) must form the number of CATs and converge to the point XC. From the continuity of Norm-2, XC 2 = INFX__ X2 is established, and from the closed one, the point XC is in C. Here, when the inequality (2. 2. 3) is executed, and when the XC is the basis of the coordinates 0, the XC is in C only at that time. Then, XC 22 = XC, XC XC, Y XC 2 Y2, the last inequality is led by the inequality of Koshi-shvartz. The distribution of each side of XC 2 means that it is XC 2 Y2 for all y∈C, but if it is XC X2 above C, any T∈ [0, 1] is an arbitrary Y. For ∈C, XC 22 ( 1-T) XC + TY22 = XC + T ( Y-XC) 22 = XC 22 + 2T XC, Y-XC + T2 Y-XC 22.

Introduction to probability optimization

If XC 22 and T2 Y-XC 22 are drawn from both sides of the above inequality, -T2 Y-XC 22 22 2T XC, Y-XC. If you divide both sides of the inequality by 2T, it will be T-Y -XC 22 XC, Y-XC 2 for all t∈ (0, 1]. You can get it.

If this theorem is given, a simple transition will provide a more general demanding characteristics to the convex gathering. KOROLLARIA 2. 2. 6 (imaging to convex gathering). At this time, there is a single point πc (x) called X-πC (x) 2 = INFY EDC X-Y2, that is, πc (x) = Argminy Edc Y-x22. This projection is characterized by inequality πc (x) -x, y-πc (x) 0.

(2. 2. 7) About all y∈Cs. If the abov e-mentioned inequality ceremony is used (2. 2. 5), 1 x n-xm 22 2m2 + 2-2m2 = 2. 2 √ Especially X N-XM 2 2; XN was optional (XN). Must forms the number of CATs, and therefore converge to point XC. From the continuity of Norm-2, XC 2 = INFX__ X2 is established, and from the closed one, the point XC is in C. Here, when the inequality (2. 2. 3) is executed, and when the XC is the basis of the coordinates 0, the XC is in C only at that time. Then, XC 22 = XC, XC XC, Y XC 2 Y2, the last inequality is led by the inequality of Koshi-shvartz. The distribution of each side of XC 2 means that it is XC 2 Y2 for all y∈C, but if it is XC X2 above C, any T∈ [0, 1] is an arbitrary Y. For ∈C, XC 22 ( 1-T) XC + TY22 = XC + T ( Y-XC) 22 = XC 22 + 2T XC, Y-XC + T2 Y-XC 22.

Introduction to probability optimization

If XC 22 and T2 Y-XC 22 are drawn from both sides of the above inequality, -T2 Y-XC 22 22 2T XC, Y-XC. If you divide both sides of the inequality by 2T, it will be T-Y -XC 22 XC, Y-XC 2 for all t∈ (0, 1]. You can get it.

If this theorem is given, a simple transition will provide a more general demanding characteristics to the convex gathering. KOROLLARIA 2. 2. 6 (imaging to convex gathering). At this time, there is a single point πc (x) called X-πc (x) 2 = INFY EDC X-Y2, that is, πc (x) = Argminy Edc Y-x22. This projection is characterized by inequality πc (x) -x, y-πc (x) 0.

(2. 2. 7) About all y∈Cs.

confirmation. This sentence can be understood when X∈C. In the case of X∈C, the result is to consider the large amount C = C-X, and the axiomes 2. 2. 2. 2. Coloralia 2. 2. 8 (no n-vulnerable imaging). The large amount of the convex mass is no n-exponential, especially πc (x) -Y2 x-y2 for any x∈RN and y∈C. At the time of X∈C, this inequality is definitely, and as a result, X∈C is actually allowed. If Y is folded inside and pulled inside, 0 πc (x) - x, y-π πc (x) -Y + y-x, y-πc (x) = -πc (x) --Y22 + Sort the formula on y-πc (x), which is y-x, sort, inequality of kosh i-shvartz or gölder, basically πc (x) -Y22 y-x, y-πc (x) y-x2 Use y-πc (x) 2. Here, the both sides are divided to πc (x) -Y2.

The nature of the projections of the projection is not only the existence, but also the fact that the large amount of convexity is all in the sem i-reality, which is included, and is actually all kinds. The fact that the large amount of convex is guaranteed to have all the possibilities destroyed by the hyperplabia. Furthermore, if either set is compact, splitting may be strict.

Figure 2. 2. 9. A larg e-capacity point x assignment of vector v = x-πc (x). DROGEN 2. 2. 10 (strictly divided). C- is a large amount of closing. Vector V is similar to each point, X∈C, actually V, X & Amp; gt; Sup V, Y.

In addition, you can arrest vector v = x-πc (x) and V, x supy edc V, Y + V22. See sketch 2. 2. 9. 9. True, because it is x∈C, x-πc (x) = 0. If v = x-πc (x) is substituted, due to the nature of (2. 2. 7), in fact, 0 v, y-πc (x) = v, y-x -πc (x) = v , Y-x + v = v, y-x + v22. In particular, for all Y∈Cs, V, X V, Y + V2 is determined.

Drying 2. 2. 2 (strict distribution of convex set). C1, C 2-One large amount of closed convex, one, one, is compact. During this time, there is a vector V similar to INF V, X & Amp; amp; gt; gt v, x.

Proof. Large amount C = C 1-C2 is closed with convex. And since VN 2 = 1, the priority (VN) belongs to a huge amount of small size. 3. If it is necessary to clarify next time, it is clear that VN actually exists, actually VN → V, and V, X V, Y exist for all Y∈Cs. Axiki 2. 2. 16 (half-spatial intersection). C RN- is a large number of balls that closed. During this time, C is considered an intersection of all space including it.

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The whole carbal position is a really compact large number in AIOGLU. However, in the weak small giant, each priority includes*some convergence, a vector, and vector V, which is actually VN (m), actually VN (M), Y → v, there is y for all Y.

) Here, another relationship: Assuming XICLEBD C HX ⊆ C.), there is a contradiction, so in fact Z∈X will fill Z∈C to our assumptions. Since C is closed, the projection from Z to C (Z) Vol Z-πc (Z), zamp & amp; gt; gt; supy edc z-πc (z), y meets 2. 2. 10. In particular, when vz = z-πc (Z) is determined, the hyperposter is considered a C carrier at the πc (Z) point (Collollary 2. 2. 6), and the hal f-space has C but has C. As a very no n-concrete consequence of 2. 2. 16, a good convex characteristics of all affin functions that minimize convex functions (that is, the Affin function, which is less than the initial function) is obtained. If F is a closed convex function, that is, if the EPI F is a closed convex function, EPI F is a intersection of all hal f-space including F. What is difficult is actually indicating that this intersection can be restricted by a no n-convex half space. See concept 2. 2. 18.

Figure 2. 2. 18. Under evaluation (dashed line) by function F (continuous blue line) and Affin. Theorem 2. 2. 19. F is a convex function that closed. During this time f (x) =

Proof First, be aware that EPI F is closed by the definition. Furthermore, it is possible to include EPI f = ∩ (here H means a hal f-space). More directly, any hal f-space can be registered via (V, A, C) ∈ RN x R x R, and HV, A, C = holds.

Because it is a. h⊃ epid, you must arrest T → φ where A becomes 0. If a is 0, that is, f (x; u) = infα & amp; amp; gt;) -F (x) In fact, x is on the side of DOM F and α & amp; gt; 0 In the case of X + αu ∈ DOM F, we pointed out that f (x; u) = + φ should be, and we proved the right axiom. Priority 2. 4. Each point X∈Dom F, there is also a conductive function of F (x; u) for each U, 1 1 f (x; u) = lim [f (x + αu). -F (x)] = infl [in f-f (x + αu) -F (x)]. In the case of α & amp; amp; gt; 0 α α ↓ 0 α x∈intom f, that is, X1 = x + Δ & amp; gt; x & amp; gt; gt = Δ = x0 Decide. During this time, x 1-x0 = 2δ and x1), f (x0) = f (x) -F (x) δ + 2δ2 f (x) f (x) = f (x) + f (x) δ + 2δ2 f (x0∈ [ x-Δ, x + δ]. CANX = f (x1) + f (x0) -2F (x) 0 (last inequality in x1, riding concave), these equations are proven. , Actually x1) + f (x0)]. CA = 2A2 [F (XI) → F (X) for Δ → 0 and Cher /2δ2 0 for all Δ & amp; gt; 0, by content in the we have f (we need to possess C X1) + f (X0 ) & amp; amp; gt; = lim sup Δ2 0. 2F (x) = Lim Sup

It will be. 2. 4. Dots X∈Dom and F-conditions and optimal conditions f are direction: (2. 4. 1)

Many functions-Subkraden

In fact, as intuitively clear, EPI F is only considered convex during this time, so many subtraders must be ∂F no n-IT, and maintain super gravity for EPI F. It is established. In other words

In other words, I need to have a larg e-scale underestimation for myself every time. When function F is convex, it is closely related to the optimal criterion of the convexity of F (when F is different, it is regarded as a larg e-scale linear under evaluation F).

F (x1) + G1, x-x1

F (x2) + g3, x-x2 f (x2) + g2, x-x2 x2

Figure 2. The upper age of the convex function. In point x1, the gradient is considered a secondary gradient G1. In point x2, there is a su b-gradient of a certain amount for reasons such as the function of the function is not different. Primary function, data G2, g3 ∈f (x2). Life and properties of subguders in which many non-ways are subdivided.

The first axiom

Papers 2. 4. 3. During this time, ∂f (x) is intact, closed convex, and compact. Claim. In fact, there is no problem that ∂f (x) is closed and convex. To believe this, actually (∂f (x) = z

This is all closed convex hal f-space intersection. Here, it is necessary to show ∂f (x) = ∅. In fact, this is derived from the following facts: many EPI Fes have a no n-Vulgar support at the point (x, f (x)). In fact, it is known that vector V and scalar B have V, X + BF (X) V, Y + BT, for all (Y, T) ∈ EPI F from the theorem 2. 2. 14. It is (that is, y and t are f (y) t). If you sort in some places, it will be V, X-Y B ( T-F (X)), and if y = x, it can be seen that B 0 Set Y = x + V. To verify that ∂f (x) is compact, F is limited to 1-Shar around X from Lemma 2. 3. 1. Since it is X∈ Nts F according to the assumption, there is a & amp; amp; gt; 0 like X + B ⊂ int Dom f for 1-shar b =. Supvavy f (x + v) = m 0; comes from Remma 2. 3. 1. In particular, if α = 1 and u = y-x, the standard Subbraduite inequality called F (x) + g, y-x f (y) is held. Therefore, if it is g∈s, it will be g∈∂f (x). The opposite is also the same, and for any g∈∂f (x), f (x + αuu) f (x) + g, x + α u-x = f (x) + α g, u. G. , UF (x; u) and the definition of a subtracker.

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Performance (2. 4. 5) proves that ∂f (x) is compact as described in the theorem 2. 4. 3. Since f (x; u) is limited to all U, all g∈∂f (x) fills g2 = since X∈ Intom F.

A more complex example is given by any vector norm. In this case, the use is that the tw o-t o-norm is defined in y ∗: =.

In addition, X = Supy: Y1 Y, X stand. If x∈Rn is fixed, g ∗ 1 is g, x = x, and ∗ x + g, y-x = x-x + g, y.

You can claim. As a more specific example, x/ x2 ∂ x2 = if x = 0.

Fig. 2. 4. 10. The point X minimizes fs above C when the g∈∂f (x), g, y-x is 0 for all y∈C ( Indicates a level curve).

This axical has the necessary and sufficient conditions to minimize convex focus f in point X, and summarizes the normal condition of the former optimal condition for the differentiated function F (for example, 4. 2. 3 B). [12] Clause). The intuition of worth 2. 4. 11 has a vector G in the large amount of ∂f (x), and is actually regarded as a hyperplabian that supports the execution of a huge amount C at point x. Is. The 10 sketch shows this behavior. Priority 2. 11. F- Convex function. Basta X∈ints, for all Y∈C, g, g. -X 0., most of the huge amount C, during this time, and actually the same side slope g∈∈. Rub the F only when ∂f (x) exists.

Proof. This axiom is not difficult. In fact, choose y∈C. During this time, g∈∂f (x) is of course present, g, y-x 0. During this time, according to the definition, it becomes f (y) f (x) + g, y, y, y, y-x f (x). This has been established for any Y∈C, which is obviously optimal. During this time, any Y∈C and any T 0 are actually X + T ( Y-X) ∈c, we have F (X + T ( Y-X)). -F (x). Here, because there is a contradiction, we will say that there is a positive set of G, Y-X 0-step for all G∈ ∂ f (x). The first motivation to select such an update is due to such a precedent, and in fact X minimizes convex focus f when 0 = ∇f (x); We believe that more persuasive inference is thrown from the idea of modeling a minimized convex function. In fact, the update (3. 2. 1) is considered as follows.

Introduction to the probability optimization lecture

It is equivalent (3. 2. 2)

1 x-xk 22. xk + 1 = argmin f (XK) + ∇f (XK), x-xk + 2AK x

This interpretation is due to the good linear list X →, which is the best linear proximity of function F in point XK.

This is the best linear proximity of function F in point XK, and we want to achieve the progress of minimizing X. As a result, we minimize this linear approximation, but we have XK's very large deletion penalty to guarantee that it is literally compatible with function F. The next contract is added, and it actually leads to an invalidation of linear approximation. Sketch 3.

f (xk) + ∇f (XK), x-xk + 12 x-xk 22 f (x)

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Figure 3. 2. 3. Left: The function f (x) = log (1 + ex) (blue) in the point xk = 0. Right: The function F (x) = Log (1 + ex) linear + 2nd upper world) at the point xk = 0), if the pitch size is ak & amp; amp; gt; 0 It is a monotonous reduction of the target F (XK). We don't spend time on the convergence of the slope method, but the choice of pitch-Ok is often very important, and we want to say that a lot of research is dedicated to the thorough choice of direction and step length; NESTEROV [44] perfectly illuminates many major problems. Subgradien t-Algorithms subgradien t-Methods is one of the (3. 2. 1) methods, and a side gradient is used instead of a gradient. This method can be easily written: K = 1, 2, ... Select an II partial gradient GK ∈ ∂f (XK) to repeat it. Step on the II. (3. 2. 4).

Unfortunately, the subcladita method is generally not a descent law. As a simple example, take function f (x) = | x x |, and x1 = 0 1] shall be as an upward direction. This is not just a coincidence that 0 is optimal for F. For example, consider F (x) = x1, and use X = E1∈RN, which is the first standard base vector. And ∂f (x) = E1 + n I = 2 Ti ei, here ti∈ [-1, 1]. N n any vector G = E1 + i = 2 Ti EI I = 2 | & amp; amp; gt; 1 is the direction of F climb, which is f (F (for all α & amp; amp; amp; gt; 0 X-αG) & amp; amp; gt; f (x). For example, if g∈∂f (e1) is uniformly selected randomly, the probability that G is downward will be smaller in ND dimension. 4, f (x+tu) -f (x), form, you f (x; u) = limt → 0 t derived in the direction, and fact that f (x; u) = SUPG ed∂f (x) G, u Ensures that arginage of (x)

This is the descent direction, but it is not proven here. In fact, in order to seek such a descent, it is necessary to clearly calculate the subclarer ∂f (x), but this is sel f-evident depending on the function, and a su b-clarady that moves with a subclader. Contrary to the simplicity of the entry method (3. 2. 4). The point X does not minimize F (X), but partial gradient reduces the distance to the boundary value: optimal point. In fact, g∈∂f (x) is set to x∈ aggmin f (x) (assuming that this is the case). Next, 1 x-α g-x 22 = x 2 2-x 2 2-α G, x-x + g22 forms for any α that is 1 α2. 2 2 2 The point is α & amp; amp; gt; 0, which is strictly smaller than 12 x-x 22, as shown now. Use F (Y) F (X) + G, Y-X for all Y, which contains the subgraphs, that is, all Y, including X. + 2 2 (3. 2. 5), regardless of the selection of g∈ ∂f (x), there is 2 (f (x) -F (x)) Mean x-α G-x 22 0. Is x-α G-x 2 2-petal. II. For all x and all g∈∂f (x), there is a subgraded edge G2 M 0). 1 K Colorarian 3. AK = K I = 1 αi, XK = ak K = 1 αk XK. Then, 2 2 x1 - x 22 + k k =

The theorem 3. 2. 8 can provide some basic convergence guarantees based on the selection of step sizes. For example, at αK → 0

In addition, you can set a specific step size to optimize the constraints. For example, suppose that R2 = x 1-x 22 is the distance (radius) to optimization. Then, select a fixed step size αk = α (3. 2. 9).

αM2 R2 +. 2kα 2 R √ is obtained.

Optimize this extreme and take α =.

Considering that the RM F (XK) -F (x) √. K Subdraduate Descent method is not a descent method, instead of tracing (tracing) the points, or using the final point, so far. It often makes sense to use the best points observed. Of course, if = argmin f (XI) xbest k xi: IKK

= F (xbest and recognisses fbest k), we have the same warranty for convergence, actually 2 R2 + k k = 1 αk M best f (XK) -F (x). K 2k = 1 αK may be several strict elections of step volume, for example, at the end of the lecture, more than our framework, detailed details of these options and applications. We will send you to recognize, because our concerns are certainly limited. Now, we are giving a sample to find the use of ROBAS data adaptatio n-statistics and other scenarios. As a motivation scenario, we are allowed to have a vector ai∈RN and the priority of the answer bi∈R, and we have some vector X's internal work AI, X. I want to do BI through. If you have a motivated answer BI or if there is other data damage, the natural goal of this task considers the data matrix A = [A 1-am] ∈ RM x N and the vector b∈ RM. , 1 1 unconditional overhadowing A X-b1 = | AI, x-bi |

In point X, K = 4000 repetition with fixing ghost volume αk α fixed to all K.

Introduction to the probability optimization lecture

Figure 3. 2. 11. When the su b-class method is used for the assignment of robot regression (3. 2. 0), the number of steps is fixed.

SJ 1-YJ (ATJ X-β),

The smaller the size of the step, the more the other (last) performance of the XK will be much slower. 3. 3. For example, Lasso [57] and compressed sound source application [20] minimizes tasks based on the similarity of A X-B22. K during this time

K R2 1 + αK M2. 2AK 2k = 1

confirmation. The starting point of the confirmation is considered to be the same preceding type as used earlier, that is, distance XK+ 1-x 22. In this case, it is noticed that projection never increases the distance to point x∈C with all probability. For example, XK+ 1-x 22 = πc (X K-αK GK) -x 22 X K-αK G K-x 22. By the way, as in the last conclusion, if the inequality is applied (3. 2. 5), α2 1 1 XK + 1-x 22 X K-x 2 2-αK [F (XK) -F (X)] + K GK 22 can be obtained. 2 2 If you divide it with 2 αK, you can actually get α 1 1 X K-x 2 2-xk + 1-x 22 + K GK 22. F (XK) -F (x) 2AK 2 By the way, from the axians 3. 2. 7, K-k A1 X K-x 2 2-xk + 1 - x 22 + K GK 22. [F (XK) -F (X)] (3 3. 7) 2AK 2 k = 1

If the average section is sorted to the section (3. 3. 7), the K 1 X K-X 2 2-xk+ 1-x 22 2AK will be obtained.

K 1 1 1 X-x 2 2-x 22 X K-x 22 + = -2AK 2A K-1 2a1 1 2AK K K = 2

K 1 1 1 2 R2 + -R 2AK 2A K-1 2a1 k = 2

Introduction to probability optimization

It is αK αK-1. In fact, it is noted that the latter sum is a telescopic, and GK 22 M2 can be obtained in inequality (3. 3. 7). √ If the number k of ITRITIONS is known, this method allows you to obtain the same strong speed Velocity as a fixed step volume, but distinguishes the warranty for all ITYETS K. In fact, we actually have k K √ 1 1 √ T-2 dt = 2k, k 0 k = 1 k 1, and as a result, if you take XK = K = 1 XK, it is appropriate. You can get a result. As a result, if xk = k = 1 xk, the appropriate result is obtained. 3. 8. In addition to the conditions set forth in the preceding paragraph, 3. 3. 6 is held. During this time F (XK) --F (X)

R2 M2 α √ + √. 2A K K

√ In this way, it can be seen that convergence is guaranteed at the “best” speed of 1/k in all repetitions. The "best" here is because of the fact that this speed is not improve d-this method has the worst case function that cannot achieve a faster convergence speed than √ RM/ K, but in fact. You can be convinced of one of the best behaviors by applying the problem structure. 3. 4. Probabilization inferior slope method The real strength of the inferior slope that has become a trivial in the last 10 to 15 years is due to the application to larg e-scale optimization issues. However, the su b-slope method only guarantees that the convergence is not rushed, and it takes half a repetition to achieve the accuracy. In fact, su b-gradient methods achieve a convergence rate that cannot be ignored due to many noise optimization issues, and is often very efficient from the viewpoint of calculation. The main structural elements of the probability optimization problem (vice) gradient method are stochastic (vice) gradient, and are often called stochastic (vice) gradient oracles. F: rn → r∪ is convex, x∈Dom f.

Here, s∈S is a choice from P.

G or G (X) is often used to reduce any vector G (x, s) unless it causes confusion to use relative notes. The normal case of this problem group is the probability planning method, and I want to solve the convex optimization problem (3. 4. 2).

Follow the x∈C and minimize f (x): = EP [f (x; s)]. < SPAN> √ In this way, it can be seen that convergence is guaranteed at the "best" speed of 1/k in all repetitions. The "best" here is because of the fact that this speed is not improve d-this method has the worst case function that cannot achieve a faster convergence speed than √ RM/ K, but in fact. You can be convinced of one of the best behaviors by applying the problem structure. 3. 4. Probabilization inferior slope method The real strength of the inferior slope that has become a trivial in the last 10 to 15 years is due to the application to larg e-scale optimization issues. However, the su b-slope method only guarantees that the convergence is not rushed, and it takes half a repetition to achieve the accuracy. In fact, su b-gradient methods achieve a convergence rate that cannot be ignored due to many noise optimization issues, and is often very efficient from the viewpoint of calculation. The main structural elements of the probability optimization problem (vice) gradient method are stochastic (vice) gradient, and are often called stochastic (vice) gradient oracles. F: rn → r∪ is convex, x∈Dom f.

SJ 1-YJ (ATJ X-β),

G or G (X) is often used to reduce any vector G (x, s) unless it causes confusion to use relative notes. The normal case of this problem group is the probability planning method, and I want to solve the convex optimization problem (3. 4. 2).

Follow the x∈C and minimize f (x): = EP [f (x; s)]. √ In this way, it can be seen that convergence is guaranteed at the “best” speed of 1/k in all repetitions. The "best" here is because of the fact that this speed is not improve d-this method has the worst case function that cannot achieve a faster convergence speed than √ RM/ K, but in fact. You can be convinced of one of the best behaviors by applying the problem structure. 3. 4. Probabilizer inferior slope method The real strength of the inferior slope that has become trivial in the last 10 to 15 years is due to the application to larg e-scale optimization issues. However, the su b-slope method only guarantees that the convergence is not rushed, and it takes half a repetition to achieve the accuracy. In fact, su b-gradient methods achieve a convergence rate that cannot be ignored due to many noise optimization issues, and is often very efficient from the viewpoint of calculation. The main structural elements of the probability optimization problem (vice) gradient method are stochastic (vice) gradient, and are often called stochastic (vice) gradient oracles. F: rn → r∪ is convex, x∈Dom f.

Here, s∈S is a choice from P.

SJ 1-YJ (ATJ X-β),

Follow the x∈C and minimize f (x): = EP [f (x; s)].<[XT a]l >Optimization algorithm for data analysis

For the natural probability gradient that requires only the calculation time of o (n) (for the o ( m-n) that calculates A X-B), the index i∈ [m] is equally selected, and G = Return AI Sign (AI, X-BI). In a more common case, when solving problems including large data sets, m 1 f (x) = f (x; si), m I = 1

Then, the random selection of index i∈ and the random g∈ ∂x f (x; si) selection are considered a probability gradient. It takes only time to calculate this probability gradient, but this is important for calculating the substance of many partial slopes ∂ f (x; si), and the normal partial gradient method used for such problems. Then, each repetition requires another M again. In general, the expected value e [f (x; s)] is generally difficult to calculate, especially when S is considered as a hig h-dimensional distribution. Especially in the application of statistics and machine learning, it is not possible to monitor distributed pandomists, and IID selection SI to P will be monitored. In such a case, it may not be possible to calculate the secondary gradient F (x) ∈ ↪ sm_2202F (x), but since it is likely to be selected from P, it is actually calculated to calculate the probable side slope. Can do.~Introduction to the probability optimization lecture < SPAN> Here is a random size in S where the distributed P (expected value EP [f (x; s)] is adopted) For s∈s, function x → f (x; s) is considered convex. During this time, if g (x, s) ∈x f (x; s), s to p as in the second lecture, and g = g (x, s), g is considered a probability gradient. You can quickly understand (remind you of the formula (2. 5. 1)). Remembering this calculation, f (y) = EP [f (y; s)] EP [f (x; s) + g (x, s), y-x] = f (x) + ep [g (g) X, S), Y-X, EP [G (x, S)], etc. are considered a probable side. Let's consider the Robust Regenerative Problem (3. 2. 10) to make the formatization (3. 4. 2) more specific.~For the natural probability gradient that requires only the calculation time of o (n) (for the o ( m-n) that calculates A X-B), the index i∈ [m] is equally selected, and G = Return AI Sign (AI, X-BI). In a more common case, when solving problems including large data sets, m 1 f (x) = f (x; si), m I = 1~Then, the random selection of index i∈ and the random g∈ ∂x f (x; si) selection are considered a probability gradient. It takes only time to calculate this probability gradient, but this is important for calculating the substance of many partial slopes ∂ f (x; si), and the normal partial gradient method used for such problems. Then, each repetition requires another M again. In general, the expected value e [f (x; s)] is generally difficult to calculate, especially when S is considered as a hig h-dimensional distribution. Especially in the application of statistics and machine learning, it is not possible to monitor distributed pandomists, and IID selection SI to P will be monitored. In such a case, it may not be possible to calculate the secondary gradient F (x) ∈ ↪ sm_2202F (x), but since it is likely to be selected from P, it is actually calculated to calculate the probable side slope. Can do.

Introduction to probability optimization here, S is a random size in the place S that has a distributed P (expected value EP [f (x; s)]), and each s∈s For, function x → f (x; s) is considered convex. During this time, if g (x, s) ∈x f (x; s), s to p as in the second lecture, and g = g (x, s), g is considered a probability gradient. You can quickly understand (remind you of the formula (2. 5. 1)). Remembering this calculation, f (y) = EP [f (y; s)] EP [f (x; s) + g (x, s), y-x] = f (x) + ep [g (g) X, S), Y-X, EP [G (x, S)], etc. are considered a probable side. Let's consider the Robust Regenerative Problem (3. 2. 10) to make the formatization (3. 4. 2) more specific.

For the natural probability gradient that requires only the calculation time of o (n) (for the o ( m-n) that calculates A X-B), the index i∈ [m] is equally selected, and G = Return AI Sign (AI, X-BI). In a more common case, when solving problems including large data sets, m 1 f (x) = f (x; si), m I = 1

Then, the random selection of index i∈ and the random g∈ ∂x f (x; si) selection are considered a probability gradient. It takes only time to calculate this probability gradient, but this is important for calculating the substance of many partial slopes ∂ f (x; si), and the normal partial gradient method used for such problems. Then, each repetition requires another M again. In general, the expected value e [f (x; s)] is generally difficult to calculate, especially when S is considered as a hig h-dimensional distribution. Especially in the application of statistics and machine learning, it is not possible to monitor distributed pandomists, and IID selection SI to P will be monitored. In such a case, it may not be possible to calculate the secondary gradient F (x) ∈ ↪ sm_2202F (x), but since it is likely to be selected from P, it is actually calculated to calculate the probable side slope. Can do.

Introduction to probability optimization

The stochastic subgraduates can overcome the probable subgraduates method (projected), taking into account this motivation. (1) E [GK | XK] ∈ ∈。。。。。。。。。。。。。。。。。。。。。。。。。。。。 (2) Execute the projected subtrasser XK+1 = πc (X K-αK GK) step. This is essentially the same as the method of the projection gradient (3. 3. 1), but the fact that the probability of transportation of the probability gradient is actually changed is different. The convergence of this method is analyzed in the appropriate clause, but here are two examples that show the normal behavior of these methods. Example 3. 4. 3 (Robe regression): Consider the task of Robate regression from equation (3. 2. 10), determine 1 | AI, X-BI | Provides X2 R, M.

Use random selection g = a i-Sign (ai, x-bi) as a probability gradient. generate a = [a1 - -am], ai ~ n (0, in × n), bi = ai, u + εi | 3, id (here, εi to n (0, 1) and u-It draw a Gaussandam size with the same coove). In this experience, n = 50, m = 100, and r = 4 are used.

Repeated K sketch 3. 5. Productivity of stochastic subtropic method in tasks (3. 4. 4) and infocate subtraofect method. Figure 3. 4. 5 shows the result of a stochastic slope method and a repetition descent of the normal imaginary su b-race lace method. < SPAN> Stochastic subgraduates can overcome the probable subgraduates method (projected), taking into account this motivation. (1) E [GK | XK] ∈ ∈。。。。。。。。。。。。。。。。。。。。。。。。。。。。 (2) Execute the projected subtrasser XK+1 = πc (X K-αK GK) step. This is essentially the same as the method of the projection gradient (3. 3. 1), but the fact that the probability of transportation of the probability gradient is actually changed is different. The convergence of this method is analyzed in the appropriate clause, but here are two examples that show the normal behavior of these methods. Example 3. 4. 3 (Robe regression): Consider the task of Robate regression from equation (3. 2. 10), determine 1 | AI, X-BI | Provides X2 R, M.

Use random selection g = a i-Sign (ai, x-bi) as a probability gradient. generate a = [a1 - -am], ai ~ n (0, in × n), bi = ai, u + εi | 3, id (here, εi to n (0, 1) and u-It draw a Gaussandam size with the same coove). In this experience, n = 50, m = 100, and r = 4 are used.

Repeated K sketch 3. 5. Productivity of stochastic subtropic method in tasks (3. 4. 4) and infocate subtraofect method. Figure 3. 4. 5 shows the result of a stochastic slope method and a repetition descent of the normal imaginary su b-race lace method. The stochastic subgraduates can overcome the probable subgraduates method (projected), taking into account this motivation. (1) E [GK | XK] ∈ ∈。。。。。。。。。。。。。。。。。。。。。。。。。。。。 (2) Execute the projected subtrasser XK+1 = πc (X K-αK GK) step. This is essentially the same as the method of the projection gradient (3. 3. 1), but the fact that the probability of transportation of the probability gradient is actually changed is different. The convergence of this method is analyzed in the appropriate clause, but here are two examples that show the normal behavior of these methods. Example 3. 4. 3 (Robe regression): Consider the task of Robate regression from equation (3. 2. 10), determine 1 | AI, X-BI | Provides X2 R, M.

SJ 1-YJ (ATJ X-β),

Repeated K sketch 3. 5. Productivity of stochastic subtropic method in tasks (3. 4. 4) and infocate subtraofect method. Figure 3. 4. 5 shows the result of a stochastic slope method and a repetition descent of the normal imaginary su b-race lace method.

The warranty of the method This diagram shows a typical operation of the probability gradient method. The initial progress to improve the goal occurs very fast, but once a low accuracy (in this case, 10-1), the method will eventually stop progress. In this figure, each repetition of the probability slope requires time for O (n), while each repetition (no n-reading) of the projection gradient requires O ( n-m), about 100 times. It is necessary to explain that it is slow. Example 3. 4. 6 (mult i-class support vector): The second example is a bit complicated. We are given a collection of 16x16 handwritten Hayaton images, and we want to classify the image presented in the form of vector a∈R256 into one of 10 digits. In the general task of the class classification of the K class, a mult i-class classification device is expressed using a matrix x = [x1 x2 - xk] ∈ RN x k. When setting a databector a∈Rn, "evaluation" is associated with class L XL and A, and (when data about images is given), find a queue X that applies a lot of correct image markers. be. (In machine learning, the weight vector W1 of WK∈RN is usually used instead of X1, ..., XK, but here X is used so as not to be confused by optimization.

LORN [K]

We express an example of a certain training in the form of steam (a, b) ∈RN x, and the bal l-class loop F (x; (a, b)) = max [1 + a, x l-xb] + L = b

Here, [t]+ = max represents a positive part. When F is fixed in X, it has f (x; (a, b)) = 0 to the pair (a, b), and the classifier presented by X has a large stock, that is, F (x; (a, b)) = 0 only when A, Xb A, XL + 1 for all L = B. В эом пыбор N = 7291 ц (AI, BI) ∈ RN ×, савае поиодession хасconreid са со сандары бidulientClus сом дом дом задач 1 f (x; (AI, bi)) п уов Means

R = 40 here, we execute a stochastic gradient down with step √ 2 1 n αk = α1 / K, and here α1 = R / m, M2 = n i = 1 AI 2 It is an approximation of). In the probability slope Oracle, the index I is once randomly selected and set to g ∂x f (x; (AI, bi)).

Introduction to probability optimization

In the normal subtraction method, a projection and downdated and downdated to indicate the su b-grade of gi ∈ ∈ ∂f (x; (x; (AI, bi))) and 1 n ting g = n i = 1 gi ∈f (x). The same strategy is used for the allocation of the step amount √ αk = α1 / K, but the step α1 = 10-j / m of 5 steps is used for j∈. The results of this experience are shown in FIG. 3. 4. 8, and the optimality (vertical axis) of the opening is canceled from the calculation of X-A-vector operators, which is normalized by n = 7291. In the figure, it can be seen that the calculation of the integer partial ∂f (x) is considered wasteful: from the viewpoint of the number of repetitions, the random method of convergence of no n-integer methods is random. Thanks to the sample, the repetition acceleration (7291 ×) has significantly emphasized the highest productivity. However, although it is not shown in the figure, this advantage is usually kept in the spectrum that selects the amount of steps, telling the advantage of the stochastic slope in the stochastic planning law (3. 4. 7). There is. ♦

Figure 3. 4. 8. Fig. 3. 4. 8. Comparison of stochastic methods and no n-time methods for tasks that minimize the average loss of hinges (3. 4. 7). The horizontal axis is a measure of time spent in some way, indicated as the number of computations of the XT AI matrix-vector work. The stochastic gradient descent method is better than an important no n-configured method. The guarantee of the convergence proceeds to discuss the stochastic subtraffe method of Vailpiece. Similar to the analysis of the projection Subtraf method, we hope that we are actually compact, and there are continuous steps that do not grow in 1 k.

K R2 1 + αk M2. 2kαk 2k K = 1 < Span> probability optimization introductory course

In the normal subtraction method, a projection and downdated and downdated to indicate the su b-grade of gi ∈ ∈ ∂f (x; (x; (AI, bi))) and 1 n ting g = n i = 1 gi ∈f (x). The same strategy is used for the allocation of the step amount √ αk = α1 / K, but the step α1 = 10-j / m of 5 steps is used for j∈. The results of this experience are shown in FIG. 3. 4. 8, and the optimality (vertical axis) of the opening is canceled from the calculation of X-A-vector operators, which is normalized by n = 7291. In the figure, it can be seen that the calculation of the integer partial ∂f (x) is considered wasteful: from the viewpoint of the number of repetitions, the random method of convergence of no n-integer methods is random. Thanks to the sample, the repetition acceleration (7291 ×) has significantly emphasized the highest productivity. However, although it is not shown in the figure, this advantage is usually kept in the spectrum that selects the amount of steps, telling the advantage of the stochastic slope in the stochastic planning law (3. 4. 7). There is. ♦

Figure 3. 4. 8. Fig. 3. 4. 8. Comparison of stochastic methods and no n-time methods for tasks that minimize the average loss of hinges (3. 4. 7). The horizontal axis is a measure of time spent in some way, indicated as the number of computations of the XT AI matrix-vector work. The stochastic gradient descent method is better than an important no n-configured method. The guarantee of the convergence proceeds to discuss the stochastic subtraffe method of Vailpiece. Like the analysis of the projection Subtraf method, we hope that we are actually compact, and there are continuous steps that do not grow in 1 k.

SJ 1-YJ (ATJ X-β),

In the normal subtraction method, a projection and downdated and downdated to indicate the su b-grade of gi ∈ ∈ ∂f (x; (x; (AI, bi))) and 1 n ting g = n i = 1 gi ∈f (x). The same strategy is used for the allocation of the step amount √ αk = α1 / K, but the step α1 = 10-j / m of 5 steps is used for j∈. The results of this experience are shown in FIG. 3. 4. 8, and the optimality (vertical axis) of the opening is canceled from the calculation of X-A-vector operators, which is normalized by n = 7291. In the figure, it can be seen that the calculation of the integer partial ∂f (x) is considered wasteful: from the viewpoint of the number of repetitions, the random method of convergence of no n-integer methods is random. Thanks to the sample, the repetition acceleration (7291 ×) has significantly emphasized the highest productivity. However, although it is not shown in the figure, this advantage is usually kept in the spectrum that selects the amount of steps, telling the advantage of the stochastic slope in the stochastic planning law (3. 4. 7). There is. ♦

Figure 3. 4. 8. Fig. 3. 4. 8. Comparison of stochastic methods and no n-time methods for tasks that minimize the average loss of hinges (3. 4. 7). The horizontal axis is a measure of time spent in some way, indicated as the number of computations of the XT AI matrix-vector work. The stochastic gradient descent method is better than an important no n-configured method. The guarantee of the convergence proceeds to discuss the stochastic subtraffe method of Vailpiece. Similar to the analysis of the projection Subtraf method, we hope that we are actually compact, and there are continuous steps that do not grow in 1 k.

K R2 1 + αk M2. 2kαk 2k K = 1

The certification analysis is very similar to the test so far, simply expanding the error XK+ 1-x 22. f (x): = e [g (x, s)] ∈ ∂f (x) is defined as a expected su b-clay dent returned by a stochastic gradient Oracle, and ORING = G K-F (XK) -- K-M As an error for su b-landing. And 1 1 x-x 22 = πc (x k-αK GK) -x 22 2 2 k + 1 2 1 x k-αk G K-x 22 2 α2 1 = x k-xk 2 2-αk GK, xk + k GK 22, 2 2 Theorem 3. 2. 7 and 3. 3. 6 proof. Here, the member αK F (XK), X K-X can be added, 1 x-x 22 2 k + 1 * + α2 1 x k-x 2 2-αk f (XK), x k-x + k GK 2 2-αk Oring, X K-x 2 α 1 X K-x 2 2-αk [F (XK) -F (X)] + K GK 2 2-αk Oring, X K-X, 2 Here, we are 1 I used the standard ball from next. Except for the error of X K-X, the proof is exactly the same as the proof of theorem 3. 3. 6. In fact, if each side of the previous display is divided into αK and the location is sorted, α 1 X K-x 2 2-xk + 1-x 22 + K GK 2 2-oring, X K-X. F (XK) -F (x) 2AK 2 Inequal formula (3. 3. 7) is gained (3. 4. 10) by summarizing this inequality.

K K K R2 1 Oring, X K-x. + ΑK GK 2 2-2AK 2k = 1

All the guarantees of the convergence after this follow this basic inequality. For this theorem, it is sufficient to take the expected value and realize the following:

* x k-x] = E [G (XK) -F (XK), X K-X | XK].

Introductory book for stochastic optimization

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Therefore, e

K R2 1 (F (XK) -F (X)) + αK M2 2AK 2

It will be.

Is obtained.

Areas 3. 4. 9 indicates that the same convergence can be achieved with expectations. In fact, this does not mean that the stochastic subtrafant method is not as bad as the no n-articurate method, but the stability of the subcladish agate method containing significant noise. In this way, the subtrafante method is quite slow, but its slowness is emphasized by the fact that it has the ability to overcome more noise. Here, some results are shown about the convergence of the stochastic slope descent law. See the appendix A. 2 for stochastic convergence mode. √ Color Rally 3. 4. 11. Areas 3. During this time, 3rm E [F (XK)] -F (X) √ 2 k. It is the same as confirming the projecting gradient, and it is only worth changing α = R/m in the extreme. We can get a convergence of a repetition probability in more cases. Μ Kororalia 3. 4. 12. When αK converges to zero instead of sum (that is, K = 1 αk = URB andk → 0), K → URB F (XK) -F (X) → 0 I will. That is, Lim Sup P (F (XK) -F (X)) = 0 for all & amp; amp; gt; 0.

The above study guarantees that it converges with the highest probability of waiting time, but may be beneficial to the highest probability in the final specimen. This can be resolved by the support of a more powerful su b-tranizing order and the support of the East Chief inequality (A. 2, A. 2. 2). Priority 3. In addition to the conditions of 3. 4. 9, G2 M is all & amp; amp; gt; 0 during this time, αK R2 RM M2 + √ + 2kαk 2 k K = 1 k = 1 k = 1 k = 1k.

There is a possibility that it is not at least 1-e -2.

John K. Dochi √R KM < SPAN> Axiki 3. 4. 9 indicates that the same convergence can be achieved with expectations. In fact, this does not mean that the stochastic subtrafant method is not as bad as the no n-articurate method, but the stability of the subcladish agate method containing significant noise. In this way, the subtrafante method is quite slow, but its slowness is emphasized by the fact that it has the ability to overcome more noise. Here, some results are shown about the convergence of the stochastic slope descent law. See the appendix A. 2 for stochastic convergence mode. √ Color Rally 3. 4. 11. Areas 3. During this time, 3rm E [F (XK)] -F (X) √ 2 k. It is the same as confirming the projecting gradient, and it is only worth changing α = R/m in the extreme. We can get a convergence of a repetition probability in more cases. Μ Kororalia 3. 4. 12. When αK converges to zero instead of sum (that is, K = 1 αk = URB andk → 0), K → URB F (XK) -F (X) → 0 I will. That is, Lim Sup P (F (XK) -F (X)) = 0 for all & amp; amp; gt; 0.

The above study guarantees that it converges with the highest probability of waiting time, but may be beneficial to the highest probability in the final specimen. This can be resolved by the support of a more powerful su b-tranizing order and the support of the East Chief inequality (A. 2, A. 2. 2). Priority 3. In addition to the conditions of 3. 4. 9, G2 M is all & amp; amp; gt; 0 during this time, αK R2 RM M2 + √ + 2kαk 2 k K = 1 k = 1 k = 1 k = 1k.

There is a possibility that it is not at least 1-e -2.

John K. Dachi √R KM Areas 3. 4. 9 indicates that the same convergence can be achieved with expectations. In fact, this does not mean that the stochastic subtrafant method is not as bad as the no n-articurate method, but the stability of the subcladish agate method containing significant noise. In this way, the subtrafante method is quite slow, but its slowness is emphasized by the fact that it has the ability to overcome more noise. Here, some results are shown about the convergence of the stochastic slope descent law. See the appendix A. 2 for stochastic convergence mode. √ Color Rally 3. 4. 11. Areas 3. During this time, 3rm E [F (XK)] -F (X) √ 2 k. It is the same as confirming the projecting gradient, and it is only worth changing α = R/m in the extreme. We can get a convergence of a repetition probability in more cases. Μ Kororalia 3. 4. 12. When αK converges to zero instead of sum (that is, K = 1 αk = URB andk → 0), K → URB F (XK) -F (X) → 0 I will. That is, Lim Sup P (F (XK) -F (X)) = 0 for all & amp; amp; gt; 0.

The above study guarantees that it converges with the highest probability of waiting time, but may be useful to guarantee that it will converge with the highest probability in the final specimen. This can be resolved by the support of a more powerful su b-tranizing order and the support of the East Chief inequality (A. 2, A. 2. 2). Priority 3. In addition to the conditions of 3. 4. 9, G2 M is all & amp; amp; gt; 0 during this time, αK R2 RM M2 + √ + 2kαk 2 k K = 1 k = 1 k = 1 k = 1k.

There is a possibility that it is not at least 1-e -2.

John K. Dachi √R KM

If δ = e-2, mr 2 log Δ1 3mr √ f (XK) -F (x) √ + k √ is a probability of at least 1-δ. In other words, the convergence of O (MR/ K) has a high probability. Before indicating the evidence itself, we will explain two examples that meet the limited conditions. Remember in the second lecture that the convex focus F m is LipshitSeva only when G2 M for all gƞƞf (x) and x∈RN. 13 requires that any of the required function F (-; S) is lipsky in the region of C. More accurately, we have a loss F (x; (a, b)) = | a, x-b |, a timid return task (3. 2. 0), we have a G2 m-state. It has ∂f (x; (a, b)) = a sign (a, x-b) only when A2 m is satisfied. f (x; (a, b)) = l = b [1 + a, x l-xb] + is expanded in the exercise B. 3 task out of the loss (3. 4. 7). The secondary calculation of 1 is performed, but here again, there is also a limited ∂ f (x; (a, b)) only when A ∈ RN is restricted.

If not, take αK = and write down.

Start from the main inequality (3. 4. 10), the main inequality of the certificate 3. 4. 9. You can see that you want to reduce the probability of standing.

You can see that you want to reduce the probability. First, x k-itational is a function of Lection, ... Oring-1, and has a conditional expected value e [Oring | ξ1, ..., oring-1] = E [Oring | XK] = 0. Furthermore, first obtain ORING 2 = G K-F (XK) 2m 2m, and then obtain | Oring, X K-X ｜ Oring 2 X K-X 2 2mr using the assumption of RESTRICTED called G2m. K = 1 Oring, X K-X is the number of mult i-densary numbers of multinagale, K T2}, XK T EX P-P 2km2 R2 K2 k2 k2 k = 1 √ to all T 0 to all T 0. The inequality (theorem A. 2. 5) of Azuma can be applied. Since the assignment t = mr k, 2 k mr 1 oring, x-xk √, the next one can be obtained. P EX P-K 2k k = 1

Is obtained.

The supervision is not f (x) -f (x) = o (). In other words, the probability gradient method is very strong in noise, and convergence is literally not affected by noise. Apart from this, the probability gradient method is often applied in an atmosphere where we cannot consider fair F, including us, because we cannot access fair F, such as statistical work, for calculation reasons. It will be done. Due to the stability and performance of this noise, the subtrafunt method has become widely widespread as an actual option for many larg e-scale database optimization tasks. Thus, the introduction of a clear optimization method, albeit more expensive, may have a limited advantage (although it is actually possible to use it!). In this chapter, many sources have received notes and advice on our work. Stanford University's Stephen Boyd EE364B Course Summary [10, 11] and POLE's "Introduction to Optimization" [47]. < SPAN> supervision is not f (x) -f (x) = o (). In other words, the probability gradient method is very strong in noise, and convergence is literally not affected by noise. Apart from this, the probability gradient method is often applied in an atmosphere where we cannot consider fair F, including us, because we cannot access fair F, such as statistical work, for calculation reasons. It will be done. Due to the stability and performance of this noise, the subtrafunt method has become widely widespread as an actual option for many larg e-scale database optimization tasks. Thus, the introduction of a clear optimization method, albeit more expensive, may have a limited advantage (although it is actually possible to use it!). In this chapter, many sources have received notes and advice on our work. Stanford University's Stephen Boyd EE364B Course Summary [10, 11] and POLE's "Introduction to Optimization" [47]. The supervision is not f (x) -f (x) = o (). In other words, the probability gradient method is very strong in noise, and convergence is literally not affected by noise. Apart from this, the probability gradient method is often applied in an atmosphere where we cannot consider fair F, including us, because we cannot access fair F, such as statistical work, for calculation reasons. It will be done. Due to the stability and performance of this noise, the subtrafunt method has become widely popular as a real option of many large database optimization tasks. In this way, the introduction of a clear optimization method, albeit more expensive, may have a limited advantage (although it is actually possible to use it!). In this chapter, many sources have received notes and advice on our work. Stanford University's Stephen Boyd EE364B Course Summary [10, 11] and POLE's "Introduction to Optimization" [47].<||A − C|| : C has rank k>4. Lecture Overview: Normal sid e-b y-line descent methods and su b-su b-descent methods projected are considered to be euclided in their essenc e-the scales that are commonly recognized in the Euclid. It depends on the measurement of the supported distance, and the update is based on Eucli d-like steps. This lecture explores a method with a more substantial metric selection, which leads to a mirror descent known as a no n-Euclid descent or a subleydal descent, and a method that combines the given tasks. Let's take a look at how to explore the geometric shape of the solved optimization problem more carefully and guarantee more cheerful convergence.

4. 1. In the first lecture, in the previous lecture, XK+1 = πc (X K-αK GK), which represents Euclid hydride, repeatedly updates the subdidence for the conclusion of the problem (2. 1. 1). I checked. The convergence of these methods is m r-diam (c) lip (f) √ f (xk) -F (x) √ = o (1), k (x) √ = o (1), k (x), k (x) K Here, R = Supxyl C X-X 2, M-Invisible Lip of F on Norm, a huge amount of C relative 2-Norm,

1 n 2 g2 = g2j. M = SUP SUP (4. 1. 1)

The guarantee of the convergence (4. 1. 1) is placed based on the scale dimensio n-the diameter C and the part of the part G are measured by 2-target. As a result, the convergence ratio depends on other scale size F and C, and the question is naturally questionable whether it can be created to take one of the best behaviors according to the difficulty and geometry. Considering this lecture, we develop a method that optimally reflects the geometric shape of optimization issues using no n-eucrid or adaptive updates. 4. 2. Method descent method The first result is spent on the method of mirror descent. This fixes the basal-subdrograd update to use a different distance measurement value instead of the secondary 2.

Figure 4. 2. 1. Bregman divergence DH (x, y). The upper upper function is H (x) = log (1 + ex), lower (linear) -linear approximation x → h (y) + ∇h (y), x-y to point y.

Introduction to probability optimization

Is obtained.

Each time, we determine the gap between the meaning of h(x) at point x and the linear approximation of h(x) taken from point y, h(y) + ∇h(y), x - y, which is nonnegative in the usual power of the first inequality of a convex function. See sketch 4. 2. 1. As a usual example, if we arrest h(x) = 12 x22, then dh(x, y) = 2 1 2. 2. An established example is the footprint of taking the list of entropy possibilities of 2 x - y n from the stochastic simplex (i. e., x 0 and h(x) = j = 1 xj log xj, which restricts x to xj n x = 1). Meanwhile, d(x, y) = hj = 1 xj log yj, the entropy divergence, or kullback and label divergence. Since the magnitude of each time step (4. 2. 2) is nonnegative and convex according to the first argument, it is naturally considered as a distance-like function. In practice, I. Calculate the gradient of ∂f (xk). II. Make the update

1 = argmin f (xk) + gk, x - xk + dh (x, xk) α k xph

1 = argmin gk, x + dh (x, xk). αk x∈C

This scheme is the Miller descent method. Thus, each differentiated convex function H distinguishes a new optimization scheme, and we often try to choose H for the best ratio of the geometry of the basis set C of constraints. In fact, we have it so that H is considered as λ-asleily convex on a relatively commonly recognized measure - λ h (y) h (x) + ∇h (x), y - x + x - y2 for all x, y∈ C. Meanwhile, our task is made bold to choose a strongly convex function H, whose diameter C is normal and small - similar to h being strongly convex (as we will see, the analog limit (4. 1. 1) is implemented). In the usual updates (3. 2. 2) and (3. 3. 2), we use the quadratic Euclidean norm to trade between the progression of linear approximations x → f (xk) + gk, x - xk, and fully trust that the approach is reasonable - we regulate the progression. As a consequence, of course, to obtain a function of the function we use to guarantee a similar picture of the regulation, we therefore need it so that the function h is 1-membered (usually reduced to a strongly convex non-truant) which goes beyond the relatively universally recognized measurement-restrictive restrictions.

Miller Descent (4. 2. 3). 4 We see that the strong convexity of H is equivalent to 1 dh(x, y)x - y2 for all x, y ∈ C. 2 Example of mirror partitioning, then, to decompose the method (4. 2. 3), we give a fixed number of examples with stochastic updates, also with inspectors, and in fact, the corresponding divergence is thought to be consistent with this tight convexity. One of the pleasing consequences of solving any kind of divergence, not just Euclidean divergence, is that it is often possible to distinguish between more frequent updates and ordinary updates. Example 4. 2. 4 (Gradient Descent is Miller Descent): Let H(x) = 12 x22. Meanwhile, ∇H (y) = y and 1 1 1 1 1 dh (x, y) = x22 - y22 - y, x - y = x22 + y22 - x, y = x - y22. Thus, substituting the update (4. 2. 3), we see that the choice of H (x) = 12 x22 resumes the usual (stochastic latent) gradient method.

1 x - xk 22. Xk + 1 = argmin gk, x + 2aK x] It cannot be denied that H can be regarded as a tight convex 2-norm for any set of constraints C. Sample 4. 2. 5 (Solving a problem with the support of the exponential gradient method): Suppose, in fact, that our limiting large quantity C = RN can be regarded as a stochastic manifestation. Updating with the usual Euclidean distance support is then a rather difficult task, but there are effective implementations [14, 23], and of course more unpretentious methods are required. With this in mind, even H(x) = N J = 1 XJ Log XJ gives the necessary quantity for a convex function, so the negative entropy is considered convex. (The derivatives of f (t) = t log t are f (t) = log t + 1 and f (t) = 1/t & amp; amp; gt; 0 at t 0.) dh (x, y) =

xj log xj - yj log yj - (log yj + 1) (xj - yj) j = 1 n

xj + 1, y - x = dkl (x || y), yj

KL-SCTION between x and y (when distributed in rn +, 1, x - y = 0 above c). This distinguishes the form of update (4. 2. 3). Consider update (4. 2. 3). Simplified specification, we want to solve the minimization problem g, x +.

Let Xj be 1, x = 1, x 0.

This is not a strict claim, and in some cases it is analytically comfortable to avoid this, but our test is easier if H is strongly convex.

Is obtained.

We would like YJ & amp; GT to in fact be; 0, but this is not considered strictly important. However, we do not factor this, we write the Lagrangian for this problem, introducing Lagrange-multipliers τ∈R for limiting equation 1, x = 1, and λ∈Rn + for inequalities x 0. During this time, we get the Lagrangian n xj xj log + τxj - λjj xj - τ. L (x, τ, λ) = g, x + yj j = 1

To minimize x and find the optimal inference format, take the conductive function with X, set them to zero, ∂l (x, τ, λ) = gj + log xj + 1-log yj + τ-τ, 0 = ∂sm_2202 ↪sm_2202 ↪sm_2202 ↪sm_2202 ↪sm_2202 ↪sm_2202 ↪sm_2202 ↪ SM_2202 02 = Yj EXP (-g j-1 - τ + λj). The latter formula distinguishes all aphrodisiac XJ, so we can, for example, can arrest λj = 0, and to kill the extremes, in fact, we are τ = log (jg (j). YJ E-GJ) -I will have 1. As you can see, we rewrite this again with the definition of a clear update in the time K for the mirror alignment method for the update Y EXP (-gi) xi = n i. j = 1 YJ EXP (-gj) Mirror Alignment Law. Is actually XK and I EXP (-ak GK, i) for any coordinates of the i-1 law. (4. 2. 6) xk+1, i = n j = 1 xk, j EXP (-ak GK, j), for example, index gradient update, and is also known as an entropy mirror descent. Later, by presenting and confirming the convergence of our preceding convergence, it indicates that the negative entropy is strongly convex to 1-norm, which means that this is applied to the convergence warranty. do. Sample 4. 2. 7 (use of P-NORMA): As a last example, a secondary P-NORM has been introduced as a function that generates a distance H. These have a good nature of timid, and even the small hugeness is limited (in the difference from the 1 x2p KL-Diverse). We are recognized in the case of 4. 2. 5). P∈ (1, 2].

If you find the function φ (x) = (p-1) ∇h (x), the calculation is calculated, function 1 and * j (y) = y2-k symbol (yj) | yj | Q-1, φj (x j (x) ) = X2-P P Q, and here, P1 + Q1 = 1 is proved to satisfy * (φ (x)) = x, that is, IP = φ-1 (like φ = с 1). In this way, the update of the mirror origin in C = RN (4. 2. 3) is converted to a more complicated (4. 2. 8).

XK+1 = ONIC (φ (XK) -αK ( P-1) GK) = (∇h) -1 (∇h (XK) -αK GK).

The second renewal configuration (4. 2. 8), that is, the one connecting the reverse reflection of the gradient (∇h) -1 is more common, that is, any H, which is strictly convex and different. It is. This is the first configuration of the update of the mirror descent (4. 2. 3), and the slope "mirror" is represented by a function that generates a distance H, and is justified, so the title of the mirror descent is justified. At least, we do not believe that the true viewpoint (4. 2. 3) is an attribute that a certain amount is easy. In fact, in this case, in this case, the side effects that limit a certain amount are more difficult, and some of them make an effective decision. For example, let's actually make C =, probable Simplex. In this case, the update by P-Norm is converted to a task that inferred 1, X = 1, x 0, and x 2 to 1 minimum V, x + x2p, and here v = αK ( p-1 ) G K-φ (XK), FFIC and φ are assigned according to the above provisions. The CARUSH-KUN-TACTKER reference test (reduced) for this issue indicates that the conclusion of this problem is actually done by finding such t∈R, n.

IP (-vj + T +) = 1, and the next number XJ = ONIC (-vj + T +).

Since the argument FFIC grows with ∈ (0) = 0, this T may be found by the accuracy of O (n log 1) while searching for tw o-section. The method of the origin of the guarantee mirror of the convergence is explained and a test of the convergence is given. In this case, a certain amount of testing is more difficult than a subtracker, projection subclade, and probable subtracker method because the distance XK+ 1-x 22 cannot be expanded. As a result, we give some tricks related to the optimization conditions of the convex optimization task, and several tricks related to the universal recognized standard and its doubl e-universal standard. We are relatively tight convex with a universal scale with a function H, and the actual doubl e-t o-norm is actually suitable for Y ∗: Sup y, x. X: x1. I want to be there.

Theorem 4. 2. 9. 9. αk & amp; amp; gt; 0-Prior to the no n-growing steps. Assuming that XK is generated by the repetition of the mirror descent (4. 2. 3). In the case of DH (X, X) R2 for all x∈Cs, the following are as follows.

Introduction to probability optimization

1 2 αK GK 2 ∗. R + αK 2k

If it is αk≡α every day, for all K∈N K

K1 α GK 2 *. DH (x, x1) + α 2 k = 1

Is obtained.

Proof we start by considering the progress achieved in one update XK, but if all conventional confirmation has begun with the Rapnov function for distance X K-X 2, instead of optimization. Use the decomposition of the number of values. Apply the inequality of the first ove r-unevennes s-that is, the definition of a subtracke r-we have F (XK) -F (X) GK, X K-X. Thinking is performed to show that XK on XK+1 is to be prepared for a small term GK, X K-X for the decision of XK+1, but XK and XK+1 are friends. It exists near a friend, for example, it does not have a huge meaning. First, add GK, XK+1 (4. 2. 11).

F (xk) -F (x) GK, xk+1 - x+GK, x k-xk+1.

Here, the necessary and sufficient conditions of the former regarding the optimality of the convex optimization issues given in the axiom 2. 4. 11. xk+1 reduces the problem (4. 2. 3), so for all x∈c. -A XK+1 0 stands. F (xk) -F (x) αk Use two tricks here: algebraic similarity with DH and inequality of fense l-yang. How to operate algebraics, in fact, ∇h (xk+1) - ∇h (xk), x-xk+1 = DH (x, xk) --DH (x, xk+1) --DH (xk+1, Get XK).

In the previous consideration, F (XK) -F (X) (4. 2. 2. 2) is substituted.

1 [DH (x, xk) -DH (x, xk+1) -DH (xk+1, xk)]+GK, X K-XK+1. αK

2 This conclusion led to the fact that DH (XK+1, XK) can cancel the ratio of GK, X K-XK+1. To see this, remember the inequality of Fengel-Dr. This inequality is basically η 1 x, y x2 + y2 * 2 2 2 2 2 for any two-to-universal measurement (--- *) and any η & amp; amp; gt; 0. It claims to be PAR. In order to guarantee this, we actually have X, Y X Y ∗, which is not changed, and that is actually not changed, and, for example Regarding a, b∈R and & amp; gt; 0, we have 0 12 (η 2 a-η -2 b) 2 = η2 a2 + 2 η 1 x y ∗ 2 x + 2 ∗ y ∗ Please note. In particular, GK, X K-XK+1

αK 1 X K-XK + 1 2. GK 2 ∗ + 2 2AK

H VOUGES DH (XK, XK+1) 12 X K-xk+1 2 strong irregular assumptions, α 1-DH (XK+1, XK)+GK, XK+1 K GK 2 *. If this is replaced with inequality (4. 2. 2. 2), it becomes 1 α [DH (x, XK) -DH (x, xk + 1)] + K GK 2 *. (4. 2. 13) f (XK) -F (x) αK 2, this inequality is inequality in the 3. 3. 3. 6 (3. 3. 7) in confirmation of the 3. 3. 6 6. (3. 3. 7) Must be similar. In fact, DH (X, XK) R2 actually, in fact, in fact, in fact, actually, if, similar to the conclusion of 3. 3. 6, the first result of this axiot Is distinguished. Regarding 2, when the step size is fixed, it is actually K.

Is obtained.

= In fact, this is considered to be the second result.

1 [d (x, x1) -DH (x, xk + 1)] + α h

Before continuing the story, let's do some observations. First of all, we focus on the fact that the whole exam is similar to a almost stochastic case. Expansion from section 3. 4 is basically not difficult, so we will overcome the most major results. Except in the case where the vector GK ∈ ∂f (XK) is actually above the Ito K ∈f (XK), the vector GK ∈f (GK | XK] ∈ff (GK | XK] ∈ff (∈] It is regarded as a probability component that satisfies XK). During this time, regarding the no n-growth priority of each step αK

Introduction to the probability optimization lecture

(ΑK here may be selected depending on G1,., GK), K αK R2 2 GK ∗ ∗ (f (XK) -F (x)))

confirmation. It is the same as the confirmation of the axiatics 4. 2. 9, but confirm the G K-vector F (XK) that satisfies E [GK | XK] = f (XK) ∈f (XK). Sketch. During this time, * * + + f (xk) -F (x), X K-x = gk, x k-x + f (XK) -GK, X K-X, similar conclusions distinguishes inequaled analog ( 4. 2. 13): F (XK) -F (x) + * 1 α [DH (X, XK) -DH (x, XK + 1)] + K GK 2 ∗ + f (XK) --GK, X K-XK x. αk 2 This inequality is generated autonomously by selecting AK. Furthermore, in Itia, + * + * E (XK) -GK, X K-X] = E [F (XK) -E [GK | XK], X K-X] = 0.

Priority 4. What distinguishes the same results as? 2. 9.

Therefore, if E [G2 ∗] M2 is limited to all stochastic accumulation, it is √ 1 k, XK = K = 1 xk, αk = R / MK, so KRM R Maxk E [GK 2 GK 2 ∗] 1 RM √ 3√ E [F (XK) -F (x)] √ + mk 2k k k k = 1 √ -12 2 k. где ы восололаи, собобо оо E [G2 ∗] m2 ∗ k = 1 k, you can give a certain guarantee of convergence in several ways. First, 4. 2. 5, exploding gradient drop. N XJ's theorem 4. 2. 16. 16. C =, and understand H (x) = j = 1 xj log 1 k negative entropy. X1 = n 1-Vector, all of which records are 1/n. During this time, if xk = k k = 1 xk, the method of the index gradient (4. 2. 6) is a fixed step α Vouching.

Klog N α GK 2WAYS. + Ka 2k K = 1

Is obtained.

If you use the inequality of kosh i-shvartz and the precedent, the preceding, the preceding, X∈C, it will be 1 1 1 1 n 2 | (x j-yj) 2 2 X-y1 = XJ XJ | =. XJ XJ J = 1 J = 1 J = 1 J = 1 = 1 2 2

That is, DKL (x || y) = DH (x, y), and H is a tight convex 1-norm above C.

Receive K DKL (x | x1) α GK 2WAYS. + α 2 k = 1

If x1 =, if DKL (x || x1) = h (x) + log n log n is 0 to x∈C. In this way, the introduction of such a thing, such as 1 k and f (XK) k k = 1 f (XK) for each K, is actually distinguished from the results. 1 n 1,

The warranty of KOROLLAR4. 2. 16 and the warranty by the normal (no n-chronic) method of the imaging subcladait (that is, the introduction of H (X) = 12 x22, such as 3. 3. 6) is extremely. It is useful. In the case of the imaging subcuradiate descent, DH (x, x) = 12 x-x22 1 for all x, x∈c = (and this distance is achieved). However, 2-Norma's dual Norm is 2, as a result, we measure the value of the goalkeeper of the goalkeeper in 2-Norm. GK, assuming that GK φ is 1 for all K, GK 2 N GK φ occurs, so the key to convergence is O (1) log n/ k in n/ log n expectation gradient. It can be used as a guarantee regular (Euclid) law. Finally, the convergence of the mirror origination method is guaranteed using a P-quota that is P∈ (1, 2]. These universal introductions include superiority, and DH actually DH If the large amount of C is compact, that is, a conditional entropy X DH (x, y) = J xj log yjj is limited, and the guarantee of the convergence is distinguished in this way. ) = x2 p-Y2 p-Sign (YJ) | YJ | YJ

N 1 1 2 2-P = XP + YP-Y-YJ | YJ | YJ

Here in the inequality, the fact that Q (P-1) = P, N J = 1 is used.

Y2-P ｜ YJ ｜ P-1 ｜ P.

YP XP = YP XP = Y2-P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P PP P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P P

1 1 1 Y2p + x2p. 2 2

Introduction to the probability optimization lecture

In many cases, h (x) = 12 x-x0 2P DH (x, y) emits x-x0 2P + y-x0 2p. As a first example, we get good results. 1 x2p, here p = 1 + korollaria 4. 2. 18. H (x) = 2 (P-1) n c ⊂. K during this time

Let's say.

K 2R21 log (2n) E2 + αK GK 2WAYS. αk 2 k = 1 k = 1 k = 1 K, especially arrested αk = R1 log (2n)/k/e and xk = k = 1 xk, R1 log (2n) √ f (XK) -F (x) 3e 3E . K.

1 X2P is considered to be a tight convex confirmation. First, H (x) = 2 (P-1) p-norm, and here is that 10 is a given constant (step size) prior distribution. In this case, axiom 4. 3. 5 will be established. Practice problem B. 4. 3 is a confirmation of this research, and in the 4. 3. 3. Survey reminds me of the actual stamp of tr (a) = n j = 1 AJJ. KOROLLARIA 4. 3. 8 (Adagrad convergence). ROUR: = Supx∈c X-x φ-Radius φ of a Large Quantity C, and issues a rigid 4. 3. 5 condition. During this time, if you select (4. 3. 7) by the method of the variable metrics, you will get k 1 2 (f (XK) -F (x)) R e [tr (hk)] + αE [tr (HK)]. 。 K = 1

Checking 4. 3. 8 will lead to some results. First, by selecting α = Ringer, 1 k, the expected guarantee of the pension 32 behind e [tr (hk)] is obtained. XK = k k = 1 xk, in the rules, if GK and J are the J-numerous ingredients observed in the route, the guarantee of convergence can be immediately obtained K1 N 2 3 2 2 Ringer E [Tr (HK) (HK). ] = Ringer. E GK, J (4. 3. 9) E [F (XK) -F (X)] 2k 2k J = 1

In addition to the face confirmation (4. 3. 9), even when accepting C =, the line (4. 3. 9) is applied to someone other than the face (for example, a column 3. 4. 11). Guaranteed by the converted stochastic slope path. Except for this, the line (4. 3. 9) cannot be distinguished. There are tasks that do not have the ability to achieve the highest convergence in stochastic optimization. Because such tasks are greatly related to geometric nuances, it usually contains data (or is often equal to zero components, that is, gradient G) data. Masu.

Is obtained.

Here, the vector ai∈M = 5000, n = 1000. This is a distinctive task in a systematized task (mechanical of support vector) accompanied by a loss on the loop. For any coordinates J∈Set, set the random signal AI and j∈ to 1/j, and in the case of unpleasant, AI, j = 0. U∈N is measured randomly, BI = Sign (AI, u) in possibility . 95, and if it is unpleasant, it is BI = -sign (AI, u). In this task, coordinates AI (therefore su b-group or

Introduction to the probability optimization lecture

The stochastic NYX of Fog Tent F) naturally has a big fluctuation. Figure 4. 3. 10 shows the behavior of Adagrad for one implementation of this task to compare with a stochastic slope descent (SGD). SGD uses pitch size √ ak = α/ k, α is the best step (based on the ultimate convergence of the method), and in Adagrad, select based on the best final convergence. Use a matrix (4. 3. 7) having α like α. This figure shows the normal behavior of Adagrad for the probability gradient in a task with at least appropriate geometry. (If you select the initial steps of Adagrad well, it often exceeds the probable slope descent method (we have indicated the "correct" shape for tasks that are expected to work perfectly. Peace, peaceful. The task, which is completely approximated in the box, is a task that waits for the Splash of Adagrad, and in uncomfortable cases, it has the ability to show that it is not underestimated than a normal subtradement. Indicates a certain extended thing: F (XK), which indicates the adversity of convergence

Figure 4. 3. 12. Comparison of Adagrad and SGD convergence in tasks (4. 3. 11). Both methods are sensitive to the selection of initial volume α of the step, but contains the highest convergence than the subcoding method for each of the initial step volume of Adagrad. The stochastic NYX of < SPAN> fog tent F) has a major fluctuation. Figure 4. 3. 10 shows the behavior of Adagrad for one implementation of this task to compare with a stochastic slope descent (SGD). SGD uses pitch size √ ak = α/ k, α is the best step (based on the ultimate convergence of the method), and in Adagrad, select based on the best final convergence. Use a matrix (4. 3. 7) having α like α. This figure shows the normal behavior of Adagrad for the probability gradient in a task with at least appropriate geometry. (If you select the initial steps of Adagrad well, it often exceeds the probable slope descent method (we have indicated the "correct" shape for tasks that are expected to work perfectly. Peace, peaceful. The task, which is completely approximated in the box, is a task that waits for the Splash of Adagrad, and in uncomfortable cases, it has the ability to show that it is not underestimated than a normal subtradement. Indicates a certain extended thing: F (XK), which indicates the adversity of convergence

Figure 4. 3. 12. Comparison of Adagrad and SGD convergence in tasks (4. 3. 11). Both methods are sensitive to the selection of initial volume α of the step, but contains the highest convergence than the subcoding method for each of the initial step volume of Adagrad. The stochastic NYX of Fog Tent F) naturally has a big fluctuation. Figure 4. 3. 10 shows the behavior of Adagrad for one implementation of this task to compare with a stochastic slope descent (SGD). SGD uses pitch size √ ak = α/ k, α is the best step (based on the ultimate convergence of the method), and in Adagrad, select based on the best final convergence. Use a matrix (4. 3. 7) having α like α. This figure shows the normal behavior of Adagrad for the probability gradient in a task with at least appropriate geometry. (If you select the initial steps of Adagrad well, it often exceeds the probable slope descent method (we have indicated the "correct" shape for tasks that are expected to work perfectly. Peace, peaceful. The task, which is completely approximated in the box, is a task that waits for the Splash of Adagrad, and in uncomfortable cases, it has the ability to show that it is not underestimated than a normal subtradement. Indicates a certain extended thing: F (XK), which indicates the adversity of convergence

Figure 4. 3. 12. Comparison of Adagrad and SGD convergence in tasks (4. 3. 11). Both methods are sensitive to the selection of initial volume α of the step, but contains the highest convergence than the subcoding method for each of the initial step volume of Adagrad.

Is obtained.

5. Guarantee of the optimality of lectures: This lecture provides the basics of the optimality of some algorithms to solve stochastic optimization issues. In particular, it shows how to prove the undercarriage underworld by the introduction of the minimax lower world and evaluating the statistical certification issues. 5. 1. The procedures and algorithms we have introduced so far have worked well in many tasks: statistical, machine learning, and probability optimization, and have given the theoretical warranty to its function. It is interesting to ask whether these algorithms can be improved and how they can be improved. In this lecture, we will develop tools to prove the optimality of the probable optimization method based on such a awareness.

Introduction to the probability optimization lecture

Minimax-Rates guarantees optimality under the minimix-optimality scheme, which works as follows: A set of possibilities and the number of cases in the performance of the procedure and the number of errors in the procedure. Measure the performance of the procedure for behavior to the most difficult (Hardest) members. And we find the best procedure with this worst number error. If you explain this more formally in the context of the task of stochastic optimization, the goal is a convex focus f with a constraint x∈C when observing only a stochastic slope (or other noise information) to F. Understand the minimization complexity. (i) Convex focus F: RN → R set F, (ii) Optimized C⊂RN closed convex set, (II) probable slope Oracle, this is a sample S space, gradient display g: rn × It consists of a probability distribution P for s × f → rn, (implicitly) S. Stocytical gradient Oracle can be requested at point X at any time and at any time. The request is to build S to P with the property of (5. 1. 1). < SPAN> 5. Lecture Typelity Guarantee: This lecture provides a foundation to prove the optimality of some algorithms to solve stochastic optimization issues. In particular, it shows how to prove the undercarriage underworld by the introduction of the minimax lower world and evaluating the statistical certification issues. 5. 1. The procedures and algorithms we have introduced so far have worked well in many tasks: statistical, machine learning, and probability optimization, and have given the theoretical warranty to its function. It is interesting to ask whether these algorithms can be improved and how they can be improved. In this lecture, we will develop tools to prove the optimality of the probable optimization method based on such a awareness.

Introduction to the probability optimization lecture

Minimax-Rates guarantees optimality under the minimix-optimality scheme, which works as follows: A set of possibilities and the number of cases in the performance of the procedure and the number of errors in the procedure. Measure the performance of the procedure for behavior to the most difficult (Hardest) members. And we find the best procedure with this worst number error. If you explain this more formally in the context of the task of stochastic optimization, the goal is a convex focus f with a constraint x∈C when observing only a stochastic slope (or other noise information) to F. Understand the minimization complexity. (i) Convex focus F: RN → R set F, (ii) Optimized C⊂RN closed convex set, (II) probable slope Oracle, this is a sample S space, gradient display g: rn × It consists of a probability distribution P for s × f → rn, (implicitly) S. Stocytical gradient Oracle can be requested at point X at any time and at any time. The request is to build S to P with the property of (5. 1. 1). 5. Guarantee of the optimality of lectures: This lecture provides the basics of the optimality of some algorithms to solve stochastic optimization issues. In particular, it shows how to prove the undercarriage underworld by the introduction of the minimax lower world and evaluating the statistical certification issues. 5. 1. The procedures and algorithms we have introduced so far have worked well in many tasks: statistical, machine learning, and probability optimization, and have given the theoretical warranty to its function. It is interesting to ask whether these algorithms can be improved and how they can be improved. In this lecture, we will develop tools to prove the optimality of the probable optimization method based on such a awareness.

Introduction to the probability optimization lecture

Minimax-Rates guarantees optimality under the minimix-optimality scheme, which works as follows: A set of possibilities and the number of cases in the performance of the procedure and the number of errors in the procedure. Measure the performance of the procedure for behavior to the most difficult (Hardest) members. And we find the best procedure with this worst number error. If you explain this more formally in the context of the task of stochastic optimization, the goal is a convex focus f with a constraint x∈C when observing only a stochastic slope (or other noise information) to F. Understand the minimization complexity. (i) Convex focus F: RN → R set F, (ii) Optimized C⊂RN closed convex set, (II) probable slope Oracle, this is a sample S space, gradient display g: rn × It consists of a probability distribution P for s × f → rn, (implicitly) S. Stocytical gradient Oracle can be requested at point X at any time and at any time. The request is to build S to P with the property of (5. 1. 1).

Depending on the script in the question, the optimization procedure is S or rudimentary to provide the value of the probability gradient G = g (X, S, F), and the task is optimized. To apply priority G (XK, SK, F), K = 1, 2, ... (I)-(III) sample was made as shown on the right. A∈Rn × N-fixing positive matrix is a specific matrix, and f-rn is the span of the convex function of the exterior of F (X) = 12 × AX-B ×. During this time, C may be at least a convex bundle, to confirm the lower world, not to actually use it in the conclusion of the task-stochastic gradient IID can be arrested.

G = ∇f (x) + lection = a x-b + Ctor for lection ~ n (0, in × n). Depending on the amount, it is the closest to the actual task, and consider the task based on the Subkrady Eight Law of Lecture 3 (3. 4. 2). In this case, the convex focus F: RN x S → R is popular, which is considered a direct function of the loss F (x; s). The task is to improve fp (x): = EP [f (x; s)], and here is the apricot of the creep P for any variable S; during this time, between the spread P and the function f∈f. Is related to. .. SK builds I. I. D. and adjusts it to distributed P (in this case, the optimization procedure does not select the point of the point. SK is optimized according to the script of the

More than the probability gradient). A similar option by the natural basis gradient, the same option is g (x, s, f) ∈f (x; s) instead of providing selection S = s. Focus when you have the ability to consider only the order of. However, it is important to note that many tasks can resume gradient G∈∂f (x; s). As an example, consider the data S = (a, b) ∈ N × logistic regression task. During this time, 1 BA, F (x; s) = Log (1 + E-Ba, X) and ∇X F (x; s) = -1 + Eba, x, E. G. (a, b) are each g∈∂. It is identified from F (x; s). In a more common case, traditional linear models in statistics have a slope scaled with more information, for example, S E-selection is usually identified with g∈∂f (x; s). Here, by setting a function F and a probability slope Oracle G, the optimization procedure is the probability of having an E [GK] ∈f (XK), which is selected for the required point x1, x2, ..., xk. Pay attention to part GK. Based on these probability gradient, the optimization procedure 'XK' evaluates the quality of the procedure by defining unnecessary loss F (X), EF (G1,., GK). -inf X∈C

Here, the expected value is taken for the su b-crader G (XI, SI, F) returned by Stock Stick Oracle, and random accidents in the selected repetition, the points of the request, x1, and ... 。 X K-optimization method. Of course, if unnecessary fair significance is only considered to the fixed function F, the obvious optimization procedure achieves an excess risk of 0, which is enough to return some X∈Argminx 8. Basically, setting a more uniform concept of risk: we want to measure a good performance for all function f∈f, but in fact. You can measure the procedure performance with the worst risk X (G1, GK). -F (x), SUP E F ('X Root)

Here, Supremum assumes a function f∈f (Subcradiate Oracle G relies implicitly at this time). The best evaluation of this metrick is emphasized by minimut risks for optimization of the probable optimization tasks of the FHER of X∈C⊂RN, which includes images (5. 1. 2).

Is obtained.

We have inferior (worst case) to the spread of f∈F and all XK of the optimization scheme by introducing the probable slope sample. The criticism of (5. 1. 2) is very pessimistic: If you take the worst case about the spread of function f∈f, it is possible to arrange very difficult tasks. Although such difficulties are not taken into account, one of the answers is to develop X, which is optimally optimized for all kinds of task F.

Introduction to the probability optimization lecture

The main alignment is a large number of methods that give the lower surface of the minimax risk (5. 1. 2). Each of them converts the maximum risk, restricts it by the Bays task (for example, [31, 33, 34]), and the highly valued performance for all possible tasks. Justify. F = INFX PH (X). During this time, for each procedure, 'X is a bottom line X) -F (V) E [FV (' X) -FV]. SUP E [F ('FLEX)

Although the lower surface is rudimentary, it functions as one of the further methods of minimax-risk reduction on the lower side. Underlines also imagine that the actual 'X-PROCEDURE has been decided. In fact, 'X' is not actually parameter. Generally, it can be assumed as an auxiliary random value U binding independent of the observed su b-carrier. During this time, of course, π (V) E [FV ('X) -FV | U] inf π (V) E [FV (' X) -FV | U = U]

In other words, there is an implementation of the same auxiliary accident at least as an average implementation. This rudimentary thing can be linked to our minimax conformity procedure 'X', and in this way, we hope that all of our optimization procedures will be determined when checking the lower world. It will be a stage.

Δ = DOPT (f0, f1) x: f1 (x) f1 + δ

Figure 5. Department Optimizer F0 and F1. Optimization of a certain function with accuracy other than δ = Dopt (f0, f1) suggests that another function is actually optimized. Aperture F (X) -F is not drawn with the minimum δ. < SPAN> We have inferior (worst) for the diffusion of f∈F and all XKs of the optimization scheme by introducing the probable slope sample. The criticism of (5. 1. 2) is very pessimistic: If you take the worst case about the spread of function f∈f, it is possible to arrange very difficult tasks. Although such difficulties are not taken into account, one of the answers is to develop X, which is optimally optimized for all kinds of task F.

Introduction to the probability optimization lecture

Is obtained.

Although the lower surface is rudimentary, it functions as one of the further methods of minimax-risk reduction on the lower side. Underlines also imagine that the actual 'X-PROCEDURE has been decided. In fact, 'X' is not actually parameter. Generally, it can be assumed as an auxiliary random value U binding independent of the observed su b-carrier. During this time, of course, π (V) E [FV ('X) -FV | U] inf π (V) E [FV (' X) -FV | U = U]

In other words, there is an implementation of the same auxiliary accident at least as an average implementation. This rudimentary thing can be linked to our minimax conformity procedure 'X', and in this way, we hope that all of our optimization procedures will be determined when checking the lower world. It will be a stage.

Δ = DOPT (f0, f1) x: f1 (x) f1 + δ

Figure 5. Department Optimizer F0 and F1. Optimization of a certain function with accuracy other than δ = Dopt (f0, f1) suggests that another function is actually optimized. Aperture F (x) -F is not drawn with the minimum δ. We have inferior (worst case) to the spread of f∈F and all XK of the optimization scheme by introducing the probable slope sample. The criticism of (5. 1. 2) is very pessimistic: If you take the worst case about the spread of function f∈f, it is possible to arrange very difficult tasks. Although such difficulties are not taken into account, one of the answers is to develop X, which is optimally optimized for all kinds of task F.

Introduction to the probability optimization lecture

The main alignment is a large number of methods that give the lower surface of the minimax risk (5. 1. 2). Each of them converts the maximum risk, restricts it by the Bays task (for example, [31, 33, 34]), and the highly valued performance for all possible tasks. Justify. F = INFX PH (X). During this time, for each procedure, 'X is a bottom line X) -F (V) E [FV (' X) -FV]. SUP E [F ('FLEX)

Although the lower surface is rudimentary, it functions as one of the further methods of minimax-risk reduction on the lower side. Underlines also imagine that the actual 'X-PROCEDURE has been decided. In fact, 'X' is not actually parameter. Generally, it can be assumed as an auxiliary random value U binding independent of the observed su b-carrier. During this time, of course, π (V) E [FV ('X) -FV | U] inf π (V) E [FV (' X) -FV | U = U]

In other words, there is an implementation of the same auxiliary accident at least as an average implementation. This rudimentary thing can be linked to our minimax conformity procedure 'X', and in this way, we hope that all of our optimization procedures will be determined when checking the lower world. It will be a stage.

Δ = DOPT (f0, f1) x: f1 (x) f1 + δ

Is obtained.

The second stage is to reduce the optimization problem to the type of statistical test [58, 62, 63] by confirming the minimax limit. In order to execute this information, we recognize the distance between functions. In fact, if the F V-function is improved from this distance, other functions cannot be completely improved. Especially, let's look at the two convex focus F0 and F1. For v∈, fv = INFX∈c FV (X). The optimization distribution between the function F0 and F1 on the huge amount C is as follows.

DOPT (F0, F1; C): = SUP δ 0:

F1 (x) f1 + δ is assumed f1 (x) f1 + Δ for f0 (x) f0 + δ containing f0 (x) f0 + δ.

In other words, if there is a bathta X, this is basically FV (x) -FV DOPT (f0, f1), and X does not have the ability to completely improve F 1-V, that is, accuracy DOPT (F0) , F1) only one of the tw o-W H-function F0 and F1 can be improved. See Figure 5. 1. 3 for the image of this value. For example, if it is f1 (x) = (x) = (x) 2 and f0 (x) = (x) = ( x-c) 2 for unchanging C = 0, DOPT (F1, F0) = C2. This DP T-division can reduce optimization to verification by legitimate tasks for verifying the hypothesis. 2. Assuming that 2. v = v is selected, the procedure is examined in the FV-functional stitch nippers in cooperation with I. I. D. SK. Oracle G (XK, SK, FV). Later, taking into account the observed su b-groups, the work was done to find out which random index V was selected. It is intuitively clear that all FVs other than DOPT (FV, FV) can be completely improved. If this can be shown, we will adjust the traditional statistical result of a good hypothesis to reduce the probability of missing when checking one or data generated on v = v. You can. In the lower world, in fact, function VLEX δ-back, WA

DOPT (fv, fv; c) a is V = V in any V, V∈V.

Is obtained.

F1 (x) = M δ | x-r | and f-1 (x) = mδ | x + r |.

Examples of these functions are shown in Figure 5. 2. 7, but it is clear that the distribution (5. 1. 4) is Dopt (F1, F-1) = Δmr. In addition, we will look at the probability of this problem, while remembering that it is necessary to configure partial gradients that satisfy E [G22] M2. G | guarantees that M is M every time. Considering this, let's actually assume δ 1 and define a stochastic gradient Oracle to expand PV and v∈ in the future (5.

Define the probability gradient oracle that diffuses to point X.

At X = VR, Oracle simply returns a random signal. Meanwhile, melto mδ Sign ( x-vr) - (-sign ( x-vr)) = mδ sign ( x-vr) ∈fv (x) e [gv (x)] = 2 for v = -1, 1. Therefore, the synthesis of the function (5. 2. 5) and the probability gradient (5. 2. 6) gives a set of functions that can be completely separated from the actual gradient.

F-1 point (F-1, F1) = MRδ X: F1 (X) F1 + MRδ

Figure 5. 2. 7. Ending function with division cap (F1, F-1) = MRδ (5. 2. 5). The second stage for checking the lower limit of the minimum minimax between the dispersion is to limit the distance between the subgradements observed by our method. This means that it is difficult to test which features to optimize, which means giving a strong lower limit. At the highest level, at 5. 2. 4, a certain κ depends on the upper line of K κ2 DKL P1K | P-1. This is a local situation that allows δ to scale up with δ to get the minimum maximum. If you have a similar tw o-on, you can simply select Δ2 = 1/2κ, which emphasizes the possibility of mistakes in the long term.

1 1 κ2 $ K K $ DKL P1 | P-1 /2 1 -. 1 - $ P 1-P -1 $ 1-TV 2

Is obtained.

In order to do this, it starts with a regular lemma (the Law of KL divergence) used when there is a potentially dependent distributed study of K. This result is considered to be a specific consequence of the Bayes standard. Renma 5. 2. 8. P (-| g1, ..., GK-1) represents the relative dispersion of the GK with K's lower grade G1, ..., GK-1. And. . GK-1. For each K∈N, P1K and P-1 G1,. GK. During this time, DKL

EPK-1 [DKL (P1 (-| G1, ..., GK-1) | | P-1 (-| G1, ..., GK-1)]. 1

Using Renma 5. 2. 8, the upper world of KL, which eats K on the probability gradient (5. 2. 6), is as follows. Between P1K and P-1 5. 2. 9. Get a K observation value in the distribution of PV from the Oracle of the probability gradient (5. 2. 6). Then, to 8 45, K 3ka2. DKL P1K | P-1 proof. Since KL-SENCTION uses the rules of the chain, only separate members give the top world. First, XK is G1,. The GK-1 function (because it can be assumed that XK is determined), so PV (-| g1,., GK-1) has a distribution (5. 2. 6) of the random number of Bernoui. Distribution, that is, 1+δ 1-Δ | DKL (p1 (-| g1,., Gk-1) || p-1 (-| g1, .., gk-1) DKL 2 2 1+δ 1 + δ 1-a = log+ log 2 1- OODE 2 1+ δ = δ δ (δ δ δ </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> </s> Δ3) -Δ3) - δ-a -2 + o (δ3) = 2a2 + o (δ4) 3A2 δ LOG 1-2 for 8 45, or DKL ( -| g1,., G K-1) || P-1 ( -| G1,., GK-1)) 3AFA for 8 45 is obtained. Summary by K completes evidence.

The theorem 5. 2. 9 and the design of a wel l-defined function (5. 2. 5) can provide the highest convergence warranty for a wide range of tasks. Theorem 5. C ⊂ R N-Convex gathering containing a radius R R ball, P is a set of distribution that generates a stochastic part of G2 m with probability 1.

Proof. Combine the Le Kama Law (Lemma 5. 2. 2. 2 (and the following theorem 5. 2. 4), our design (5. 2. 5), and their probability noder (5. 2. 6).

1 AMR 1-D P ｜ MK (C, F) 2 2 kl 1-1- Next, the theorem 5. 2. 9, this is further 3KA2 AMR MK (C, F) 1-, 2 for all δ 45 2. Guarantee theactual. Finally, Δ2 =

If you select 1 k 0 for each coordinates, the lower world of minimact risk will be as follows. Proposal 5. 3. In a certain humming measurement of a certain Δ∈R+, it is assumed that V = N is a variety ⊂ F, which is δ-Seal, and to satisfy Remma 5. 3. 2. At this time, n

1 N DKL P+J | p-j. Mk (C, f) Δ 1-2n J = 1

The certificate 5. 3. 2 guarantees MK (C, F).

N $ δ (1 -$ P+ j-p -j $ TV). 2 J = 1

If you apply a Koshishbalz inequality, it will be $ $ 2 in the inequality of the Pininsker.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

If this is assigned to the previous extreme, you will get desirable results.

With the support of this proposal, we can provide some minimum lower worlds. We focus on two special cases, indicating that the probability gradient procedure we have created is the best for all sets of tasks. We give one result, and other results are on exercises related to lecture notebooks. For our main results, we use the Assius method to consider a optimization task with a large amount of C⊂RN with a radius R & amp; amp; radius. We hope that Oracle, a stochastic gradient, will satisfy the first [g (x, s, s, f) 21] M2. In other words, ｜ f (x) -f (y) ｜ m x-jyw ｜. Priority 5. 4. F and stochastic gradient Oracl e-In fact, C ⊃ [-r, r] n is actually higher.

1 N, √ mk (C, f) RM min 5 96 K confirm. Our confirmation is the same as the construction of the lower world in the past, but the minimum of the convergence rate is larger with the number of dimensions, as it is necessary to build a unique function for RN. Solve a & amp; amp; gt so far; 0. Determine the function M δ X-RV1 for each v ∈ v = n. FV (X): = N During this time, as the test shows, the set is considered a paragraph in MRA N-Humming Measurement, for example, Melty Melt FV (X) = | R 1-Sign (XJ) = VJ.

Introduction to the probability optimization lecture

Here, it is necessary to build a stochastic components (as in the previous time). Here, E1,. An d-n is the normal basal vector. Recognize the probable ingredients with the possibility of 1+δ Me j-SIGN (X J-RVJ) 2n for each v∈V (5. 3. 5) g (x, fv) = 1-tur that ME J-SIGN (X J-RVJ) 2n possibilities. In other words, Oracle selects coordinates J∈ randomly, throws coins according to this choice, and returns the j-part that was correctly signed by the possibility of 1+δ 2, and returned Me j-Sign (X J-RVJ). In that case, return the negative. If the sign (x) is a vector of the sign X, during this time, N 1+δ 1-turning Melto E [G (x, fv)] = m-code (X J-RVJ) = Certification ( X-RV). EJ N N J J = 1

That is, it is E [g (x, fv)] ∈ ∂fv (x). 5. 3 is possible here, and it is actually ⎞ n n

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

There is still to limit the members of the above KL-Diverse. The PVK means that this method means dispersion of the sodgodent that explores the FV function, and v (± J) means vector V that is forcibly equal to the J-numbered substance to ± 1. 。

1 DKL PVK (+J) | PVK (-j). DKL P+J ｜ P-J N 2 Vusion K Kk Kk

Therefore, when V and V are released with only one coordinate (the first coordinates without overhead), the DKL PV | PV is limited. M = 1 to make the calculation easier. For example, only the help of subgraph dispersion (5. 3. 5) changes, and the divergence does not change. If you use the chain law (Lemma 5. 2. 8), you can get K EPV [DKL (- | G1: K-1) || PV (- | G1: K-1)]. DKL PVK ｜ pvk = k = 1

We consider one of the standards and pay attention to the fact that K-application XK is considered as a G1, ... function. G K-1. We are DKL (PV ( -| XK) || PV ( -| XK)) PV (g = E1 | XK) PV (g = -E1 | XK) + pv (g = -E1 | XK) Log, PV (g = E1 | XK) PV (g = -E1 | XK), and PV and PV have a monotonous probability in all subguda, so in the case of g∈. Not done. If you continue inference, it will be as follows = pv (g = e1 | XK) log

DKL (- | XK) || PV (- | XK)) =

1+δ 1-OODE 1-OOZE δ 1+δ 1+Δ 1+δ log = log = log.

4) Note that this final size is restricted to 3AF n, but in the case of δ, 3KA2 4 is actually obtained. DKL PVK ｜ PVK N 5 If you replace the calculation before the lowest line (5. 3. 6), ⎞ N 2 MRA 1 3ka 3ka MRA ⎝ = 1-1-1-1-1- will be obtained. MK (C, F) 2 2n N 2n J = 1

N selection Δ2 = min distinctions on the results of axiot.

Several suggestions must be solved. First of all, this axiom resumes the result of the axiom 5. 2. 10 approves n = 1. Second, if we are angry, in fact, if we want to improve with a huge amount of 2-A or higher balls, at least if it is worse, we need to pretend to depend on the dimension. And in the end, the result is considered sharp again. Appropriate results can be obtained by worth 3. 4. 9. Coloralia 5. 3. 7. In addition to the conditions of 5. 3. 4., C⊂RN has μ-Drawe r-rinner and is in the mission of the router radius. During this time

√ √ √ √ √ √ √ mk (c, f) Router m min 1, √. Wald [59] includes a long situation in Wald [59] in 1939. Optimal information and theoretical tuning are widely developed by iBRAGIMOV and khasminsky [31], which is our tuning. Our tuning in this chapter is implemented in the [1] method of Agarval. It is implemented in the Agarval [1] method to confirm the lower world of the probable optimization task, but our results have some sharp constants. It should be noted that in our method, the fact that FAN O-Inequality has not been introduced to the lower world, which is usually to confirm the current description of the result of realization in Inf a-Doctrine [17, 63]. Used for. Recent discussions on the confirmation of the lower world in statistics are in TSYBAKOV's book [58] and Lecture Notes [21]. Being interested in a stochastic optimization task allows you to migrate from optimization to statistical test tasks. In fact, there is a book called Nemirovsky and Yudin [41] for the lower world of stochastic tasks (he still distinguishes probable optimization and statistical optimization).

Introduction to probability optimization

A. Technical application A. 1. Continuousness of convex functions This application confirms the main results of continuity of convex functions. Our discussion is based on the discussion of Hiriata-Arruti and Lemareshal [27]. Confirmation of Remma 2 can be included as x∈B1 as an EI EI. Here, E I-N is a regular basis vector, I = 1 | 1. In this way, f (x) = f is obtained.

| Xi ｜ Code (XI) EI + ( 1-X1) 0

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

Maximum < Span> √ n 1 RINNER MIN, √ √ mk (C, f) Router M min 1, √. Subsequent reading includes a long situation in Wald [59] in 1939 in Wald [59]. Optimal information and theoretical tuning are widely developed by iBRAGIMOV and khasminsky [31], which is our tuning. Our tuning in this chapter is implemented in the [1] method of Agarval. It is implemented in the Agarval [1] method to confirm the lower world of the probable optimization task, but our results have some sharp constants. It should be noted that in our method, the fact that FAN O-Inequality has not been introduced to the lower world, which is usually to confirm the current description of the result of realization in Inf a-Doctrine [17, 63]. Used for. Recent discussions on the confirmation of the lower world in statistics are in TSYBAKOV's book [58] and Lecture Notes [21]. Being interested in a stochastic optimization task allows you to migrate from optimization to statistical test tasks. In fact, there is a book called Nemirovsky and Yudin [41] for the lower world of stochastic tasks (he still distinguishes probable optimization and statistical optimization).

Introduction to probability optimization

A. Technical application A. 1. Continuousness of convex functions This application confirms the main results of continuity of convex functions. Our discussion is based on the discussion of Hiriata-Arruti and Lemareshal [27]. Confirmation of Remma 2 can be included as x∈B1 as an EI EI. Here, E I-N is a regular basis vector, I = 1 | 1. In this way, f (x) = f is obtained.

| Xi ｜ Code (XI) EI + ( 1-X1) 0

| Xi ｜ F (Code (XI) EI) + ( 1-X1) f (0)

Maximum √ √ √ n 1 1 rinner m min, √ √ mk (c, f) Router m min 1, √. Wald [59] includes a long situation in 1939 in Wald [59]. Optimal information and theoretical tuning are widely developed by iBRAGIMOV and khasminsky [31], which is our tuning. Our tuning in this chapter is implemented in the [1] method of Agarval. It is implemented in the Agarval [1] method to confirm the lower world of the probable optimization task, but our results have some sharp constants. It should be noted that in our method, the fact that FAN O-Inequality has not been introduced to the lower world, which is usually to confirm the current description of the result of realization in Inf a-Doctrine [17, 63]. Used for. Recent discussions on the confirmation of the lower world in statistics are in TSYBAKOV's book [58] and Lecture Notes [21]. Being interested in a stochastic optimization task allows you to migrate from optimization to statistical test tasks. In fact, there is a book called Nemirovsky and Yudin [41] for the lower world of stochastic tasks (he still distinguishes probable optimization and statistical optimization).

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

A. Technical application A. 1. Continuousness of convex functions This application confirms the main results of continuity of convex functions. Our discussion is based on the discussion of Hiriata-Arruti and Lemareshal [27]. Confirmation of Remma 2 can be included as x∈B1 as an EI EI. Here, E I-N is a regular basis vector, I = 1 | 1. In this way, f (x) = f is obtained.

| Xi ｜ Code (XI) EI + ( 1-X1) 0

| Xi ｜ F (Code (XI) EI) + ( 1-X1) f (0)

maximum

. The first inequality exploits the fact that | and (1 - x1) form a convex combination since x∈B1, and similarly for the second inequality. For the lower bound, x∈Int b1 satisfies x∈Intom f, so by Theorem 2. 4. 3, ∂f (x) = ∅. In particular, there exists a vector G such that for all y, f (y) f (x) + g, y - x, and further f (y) f (x) + inf g, y - x f (x) - 2 ginger ytherb1.

For all y∈B1. Proof of Theorem 2. 3. 2 First, suppose that for every point x0 ∈ C, there is an open ball b ⊂ int dom f such that $ $ (a. 1. 1) | L $ x - x $ 2 for all x, x ∈ B. . . BK with the corresponding constants for the lips L1, . . . L = Maxi Li gives the result. Therefore, we need to show that at any point x0 ∈ C, it is possible to construct a ball that satisfies the lips condition (A. 1 . 1). 3. 1 For any point x0, & amp; amp; gt; 0 and --er 0, f - be a convex ball and let b = . Let f (x) ∈ [m, m] for all x∈X0 + 2 b. ) | 2

Proof Let x, x∈X0＋B.

x - x∈X0 + 2 B, x - x2

We can see that x lies on the line segment between x and x by constructing (x-x)/x-x2∈B. Clearly, if x - x2 1 + x + x. x or x = x = x = x + x - x2 x - x2 x - x2 + x - x2 + then f (x) or

x - x 2 f (x) + f (x), x - x 2 + x - x 2 +

x - x 2 x - x 2) - f (x) f (x [m - m] x - x 2 + x - x 2 + $ m - m $ x - x $ $. 2 Roles of x and X, we obtain the result.

A. 2. Probability Background In this section we very briefly state various necessary definitions and results that we will use. Theoretical and measurement interpretations are omitted as they are not essential for our main purpose. Definition A. 2. 1. The ordering of x1, x2, ... converges with probability. The probability of a random vector x converges to prima if and only if lim sup p (xn - x care & amp; amp; gt;) = 0 for all & amp; amp; gt; 0.

Definition A. 2. In line with the probability vector, X1, X2, .... There is a set of (this can include all information about x1, x2, ...), and for any N, (i) XN is a Zn function, (II) Zn-1 is Zn. It is a function of (III), and if you have a conditional expected value condition E [XN | ZN-1] = XN-1, it is a multinager. If the conditions (i) are satisfied, XN says that it fits Z. Here, we provide a proof of Azuma-hoeffishing inequality. The first result is an important intermediate result. Lemma A. 2. 3 (Lemma Hoeffishing [30]). 2 For λ ( B-A) 2 for all λR, E [EXP (λ ( x-E [X]))] EXP 8 proof. First, note that if y is a probability variable of y∈ [c1, c2], it is ( C-C) 2 var (Y) 2 4 1. In fact, var (y) = E [( y-E [y])] 2 and E [Y] are held.

Introduction to the probability optimization lecture

Is C2 + C1 2 (C 2-C1) 2 C2 + C1 2. C2 - = (A. 2. 4) Var (Y) E Y-2 2 4 Assume 0, 0∈ [a, b]. ψ (λ) = Log E [EλX]. Then, ψ (λ) =

E [Xeλx] E [X2 EλX] E [xeλx] 2 (λ) =.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

Here, p (y) = 0 for y∈ [a, b]. Then, find e [y] = ψ (λ) and var (y) = e [y 2] -E [y] 2 = λ (λ). However, of course, the distribution P of X is supported on [a, b], so you can see that it is y∈ [a, b]. Using Taylor's theorem, ψ (λ) = var (y) stands.

λ2 λ2 ψ (λ) = ψ (0) + ψ (0) λ + ψ (λ) ψ (λ) = ψ (0) + ψ (λ) 2 2 = 0

It will be. However, ψ (λ) is desirable.

Theorem A. 2. 5 (inequal formula of Azuma-hoeffishing [4]). X1, X2, ... For all I = 1, 2, ..., the mult i-in garlic difference class with | Xi | B is B. Then, N 2T2 XI T EX P-2 P Nb I = 1

Proof. The tail below is the same, so prove the upper tail. This proof is almost a direct consequence of the ulimal method of Hoeffishing (Lemma A. 2. 3) and the extreme method of Chernoff. In fact, N N EXP (-λt) XI T EXP λ XI P I = 1 5 WE is established.

Indeed, it is possible to imagine a governing basis measure μ in equations where P contains density p.

Now, let Zi be the order in which Xi fits, and repeat the relative expectation: n-1 n = E exp λ Xi Xi exp(λXn ) | Zn-1 E exp λi=1.

X1, . Xn-1 can be thought of as functions of Zn-1. Using this calculation repeatedly, we arrive at n 2 2 λ nB . exp Xi (A. 2. 6) E exp λ 8 i=1.

Now we optimize by choosing λ 0 to minimize the upper bound highlighted by the inequality (A. 2. 6) and simply write n 2 2t2 λ nB - λt = exp - 2 P Xi t inf exp 8 nB λ0 i=1

A. 3. Preliminary results on divergence We present some usual results on divergence without confirmation, referring to the usual references (e. g. the book by Cover and Thomas [17] or the extensive work on divergence measures by Lise and Wajda [35]). Finally, we show and prove some results. The first of these is known as the data processing inequality, which basically says that manipulating a random variable (even adding noise to it) can only reduce its variance. See Cover and Thomas [17] or Liese and Wajda [35] for confirmation, axiom 14. 1 (data processing). Let P0 and P1 be the variances on any variable S∈S, Q(- | s) be the variance of any relative probability depending on s, and define Qv (A) = Q(A | s)dPv (s) for v = 0, 1 and all sets A. If we perform operations on random variables S ∼ P, we have less "information" about the initial distribution of P than if we had performed few operations. This is the consequence of Pinsker's inequality. Theorem A. 3. 2 (Pinsker's inequality). Let P and Q be random distributions. During this time, 1 P - Q2TV Dkl (P||Q) . 2

Introduction to stochastic optimization

Proof We first note that the proof of the result under the assumption that the subspace S to which P and Q are assigned is finite gives us the collective result. Suppose that for A⊂S, we obtain the advantage P - QTV = sup |P(A) - Q(A)|. A ⊂ S

(Q q is defined as tw o-price dispersion of p (0) p = p (a), p (1) = 1-p (a), and as in Q, it becomes p-qtv = p-qtv, and a. The 3. 1 sentence is actually | Q | DKL (P || Q)).

Practice questions for lectures 3 Questions B. 3. 1: In this problem and correct answer, (probable) subtr a-method (probability) Subtr a-method to study a classification device (classification device that determines numbers). Perform experiments using. Warning: Here, for example, a normal machine for machine learning and statistical research, such as 3. 4. 6, is used.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

A mult i-class classification device that supports RD x K (there is a K class here) is provided, and the prediction class for the databector A∈ RD is ARGMAX A,, ARGMAX A, XL = Argmax

Loan [K]

Here, A is a data bathta (code), and B is the data point. Loss F (x; (a, b)) = max [1 + a, x l-xb] + l = B mult i-class function.

Here, [t]+ = max means positive ratio. By applying a stochastic gradient, I want to minimize F (x): = EP [f (x; (a, b))] = f (x; (a, b)) DP (a, b). (A) Indicates that it is actually high speed. (b) When the classifier represented by X contains a huge dispatch, that is, when all L = B contains A, XB A, XL + 1, only during this time f (x; A, b) Show the method of calculating the vector g∈∂ (x; (a, b)) for (c) 0 (A, b). Note that it is g∈rd x k): This task, especially the stochastic subordinate, to determine the postal service data. Experiment to study the performance of the lugs (these data is quoted from Yanna Le Kunna's book [24], Student Zip. Test and Zip. INF INFORMATION FIL E-can be downloaded from the archived tar file http: // web. Stanford. Edu/.

Jduchi/PCMiconvex/zipcodes. tgz. Start code for Julia and Matlab is available in future addresses. In the case of julia: http: // web. Stanford.

JDUCHI/PCMICONVEX/SGD. JL II. For Matlab: http: // web. Stanford. Edu/

Jduchi/PCMiconvex/MATLAB. TGZ two methods (SGD method and Multiclassvmsubgradient method) are stored in the start code. Implement these methods (you need a mult i-class SVM slope single test code to double test). In the SGD method, the step value √ proportion αI ∝ is 1/ i, and X is projected to the ball X2IJ using a flobeniusnorm. Br: = here x2fr = IJY

We have sold a preliminary process step that still converts data that represents 1 a-a 22. During this time, a correlation is executed. Function K (a, a) = EXP (-2τ).

Convert to a vector.

φ (a) = k (a, ai1) k (a, ai2) - k (a, aim), here is I1, ... IM is a random part (see getkernelRepRESMENT). Use SGD and MulticlassvMSubGradient methods (Julia/Matlab) that sells methods. What performance can be obtained during the classification? Which is the faster of your classifier? Question B. 3. 3: This task gives a simple restriction on the convergence rate of probability optimization that minimizes strict convex functions. If c is a large amount of small convexity and F is relative 2-λ -sympathetic function on the target, λ f (y) f (x) + g, y-x + x-y22 is all G. Concerose ∈. ∂f (x), x, y∈c. 2 Here, explain the probability gradient method: A partial gradient GK with a lot of noise with I. E [GK | XK] ∈f (XK) in the repetition K. obtain. Execute the projected sub gradient step XK+1 = πc (X K-αK GK). E [GK 22

For all K, we will show whether there is a guarantee of the convergence K M2 (log K + 1) during this time to select Step volume αk =. (F (XK) -F (x *)) e 2λ M2

Lecture exercises 4. 1: In the lecture, we evaluated the descent,

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

Here, we actually justify the results of 4. 3. Using this inequality using this inequality. Justify using 3. Choose the size of the AK step adaptively in the K-M step and optimize the current convergence to the current itelato.

K-1 2 Gi 2 ∗, αk = R i = 1

Based on the previous subguder. In this case, prove that K1 K 2 2 ∗ GK ∗ (F (XK) -F (X)) 3re E K = 1 holds.

Introduction to the probability optimization lecture

Conclusion from the survey 4. 3. Hint: Correct inequality affirmation is considered to have the required ability: each no n-litter priority A1, A2, ............ K is K's K. Includes AI 2 AI. I = 1 i = 1 j = 1 AJ induction method is considered to be one of the natural strategies. Question B. 4. 2 (strong uneven P-norm): Example 4. 2. 2. 7. 1 x2p, show in Fact, h that tighry consVEX. H (x) = 2 (P-1) convex relative P-p-norm. ψ (t) = 2 (P-1) j = 1 φ (XJ)). For x∈Rn, we decide ∇2 h (W) x2p, and here we determine ∇ ∇ (W) = [φ (W1) to φ (WN)], and we are ∇. 2 h (W) = ψ

φ (WJ) DIAG φ (W1), ... φ (WN).

Here, in order to show the strong bump of H (x) = J XJ Log XJ, the actual discussion is applied in a 4. 2. 5 case, but the inequity of Hölder is used instead of the inequality of Koshi-shvartz. Question B. 4. 3 (Variation-Metric Methods and Adagrad): Remove the coming Variement-Metric method to minimize convex focus F in most subordinate.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

KK 1 1 2 2 2 2 2 2 GK H-1. X k-X k-x k-x k-1 + x 1-x H1 + E 2 k k = 2

In a diagonal matrix, the inclusion is represented by a square root from the tw o-piercing of the gradient coordinates. (This is the method of Adagrad.) Actually, x k-x 2h k-x k-x 2h k-1 x k-x mm tr (H K-H K-1), here trth (a) = n i = 1 AII Indicates that it is the release of (b) Rip = Supx∈c X-x φ is limited. At any choice from the HK diagonal matrix, we actually show that we will obtain K-K 1 1 GK 2H-1. (F (XK) -F (X)) Ringer E [TR (HK)] + E 2 K K = 1

(C) GK and J are the J-number coordinates of the sub grid of the K-number. Choose how HK is specified. Indicates that it is actually K 1 nk 2 3 2 2. (F (XK) -F (x)) Rip E GK, J E 2 K = 1

(D) Assuming that the area is C =. What is Adagrad's expectation regret? In fact, the regrets expected (with the accuracy of the numerical immutable coefficient we ignore) is the same time of the expected regrets of the normal projecting gradient, which is exactly the same as the 1k K 2 2 GK 2. show. (F (XK) -F (x)) o (1) SUP X-X 2 E Xorn

Tip: Use the koshi-shvartz method. (E) As shown in the last sub check, actually C =. This part is actually GK∈N for all K, and in fact, P (GK, J = 0) = PJ for any coordinates J. The discussion of Adagrad indicates that it is √ nk 3 k √ (f (XK) -F (x)) PJ. E 2 K = 1

What is the extreme responsibility for normal imaging gradient? Is there any other person with the possibility of Adagrad?

Practice in the fifth lecture B. 5. 1: In this task, check the bottom of a strong and convex optimization task. In fact, it is supposed that a partial lattice GK with a lot of noise that satisfies IID is obtained in each repetition of the optimization procedure.

GK = ∇f (XK) + oring, oring ~ n (0, σ2).

Introduction to probability optimization

To justify the bottom of the optimization procedure, use the function of λ fv (x) = ( x-vaf) 2, v∈. FV on R on 2 v = ± 1 the small significance of the function is FV = 0. (a) Division of two functions F1 and F-1, specifically (5. 1. 4), DOPT (F-1, F1; C): = f1 (x) f1 + Δ Includes f- 1 (x) f-1 + δ SUP δ 0: Regarding any X∈C. F-1 (X) F-1 + δ is actually λ DOPT (F-1, F1; 2, 2) when C = R (or more, between C⊃ δ, δ, δ). (B) is that the mismatch between the two conventional screams p1 = n (μ1, σ2) and the p2 = n (μ2, σ2) is exactly the same (μ1-μ2) 2 (μ1-μ2). Show 2σ2 (C) DKL (P1 | P-1) = for each optimization of each optimization using the LE KAMA method.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

Comparing the results with the upper limit of regrets of task B 3. 3. 3. Hint: When PvK means separating the raw gradient of the F V-function, it indicates that it is actually 2 kλ2 Δ2 k. DKL P1K ｜ P-1 σ2 problem B. 5. 2: C =, stochastic gradient Oracle G: rn × f → n, condition p for any coordinates j = 1, 2, ... (GJ (x, s, f) = 0) See the set of function F that satisfies PJ. In fact, if you show the maximum K∈N, the minimax bottom line for the probable oracle provided by this class is 1 √ pj, mk (c, f) c √ k j = 1 n.

Here C & amp; amp; gt; 0-numerical constant. How is this correlated with the key of the convergence selected by Adagrad?

(1) A. Agarwal, P. L. Bartlett, P. Ravikumar, M. J. Wainwright, Information-theoretic lower bound of oracle complexity for stochastic convex optimization, IEEE Trans. Inform. Theory 58 (2012), no. 5, 3235-3249 , doi 10. 2011. 2182178. MR2952543 ← 171 [2] P. Assouad, Deux Remarques Sur l'estimation (French, with English summary), C. R. Acad. Sci. Paris Sér. , 1021-1024. MR777600 ← 167 [3] P. Auer, N. Cesa-Bianchi and C. Gentile, Adaptive and confident online learning algorithms, J. Comput. System Sci. 64 (2002), no. 1, 48-75, doi 10. 2001. 1795. Colt 2000 (in English). MR1896142 ← 157 [4] Azuma, K., Weighted sum of dependent random variables, Tohoku Mathematical Society. (2)19(1967), 357- 367, doi 10. 2748/TMJ/1178243286. MR0221571 ← 174 [5] A. Beck and M. Teboulle, Mirror Descent and no n-linear projected subgradien t-methods for convex optimisation, Oper . Mirror Descent and Nonlinear Projected Gradient Methods for Convex Optimization, Oper. Lett. 31 (2003), no. 3, 167-175, doi 10. 1016/s0167-6377 (02) 00231-6. MR1967286 ← 157 [6] A. Ben-Tal, L. El Ghaoui and A. Nemirovski, Robust Optimisation, Princeton Series in Applied Mathematics, Princeton University Press, Princeton, NJ, 2009. MR0329725 ← 120 [8] D. P. Bertsekas, Convex Optimization The

151 [15] S. Bubeck and N. R Cesa-Bianchi, Regret analysis of stochastic and non-stochastic multi-armed bandit problems, Foundations and trends in machine learning 5 (2012), no. 1, 1-122. ← 157 [16] N. Cesa-Bianchi, A. Conconi and C. Genti. (1)学習アルゴリズムの汎化能力について, IEEE Trans. Inform. Theory 50 (2004), no. 9, 2050-2057, doi 10. 2004. 833339. MR2097190 ← 140 [17] T. M. Cover and J. A. Thomas, Elements of Information Theory, 2nd ed., Wiley-Interscience [John Wiley & amp; amp; amp; Sons], Hoboken, NJ, 2006. MR2239987 ← 171, 175 [18] A. Defazio, F. Bach and S. Lacoste-Julien, Saga: a fast incremental gradient method with support for non-strongly convex composite targets, Advances, Advances In neural information processing systems 27, 2014. Statist. 18 (1990), no. 3, 1416-1437, doi 10. 1214/AOS/1176347758. MR1062717 ← 167 [20] D. L. Donoho, Compressed Sensing, IEEE Trans. Inform. Theory 52 (2006), no. 4, 1289-1306, doi 10. 2006. 871582. MR2241189 ← 129 [21] J. C. Duchi, Stats311/EE377: Information theory and statistics, 2015. ← 171 [22] J. Duchi, E. Hazan and Y. Singer, Adaptive Subgradien t-methods for online learning and stochastic optimisation, J. Mach. Learning. Res. 12 (2011), 2121-2159. MR2825422 ← 153, 157 [23] J. C. Duchi, S. Shale v-Shwartz, Y. Singer and T. Chandra, Efficient projections

[25] E. Hazan, The Convex Optimisation Approach to Regret Minimisation, Optimisation for Machine Learning, 2012. ← 140 [26] E. Hazan, IntRuDuction to Online Convex Opti MISATION, Foundations and Trends in Optimissation 2 (2016), NO. 3-4, 157-325. ← 102 [27] J. Hiriart-Marty and C. Lemaréchal, Convex Analysis and Minimization Algorithms I, Springer, New York, 1993. ← Conversion Analysis と ア ア ア ア ア ゴ リ ゴ リ ゴ リ ゴ ゴ ア ア ア ア ア ア ア Ringer, New York, 1993. (1) Confmentation ア minimum ア ル ゴ リ ズム, Springer, ニュ ー ヨ ー ク, 1993 ← 101, 122 [29] J.-B. Hiriart-ADRRUTY and C. Lemaréchal, Fundamentals of Convex Analysis, Springer, 2001. 析 の Basic, シュプ リ ン 200 ー, 2001 ← 122 [30] W. Hoeffding, Probability inquality for Sums of Bounded Random Variables, J. Amer. Statist. 58 (1963), 13-30. t. A. рINT. хасминский˘ı з. Descent USing Predictive Variance Reduction, Advances in Neural Information Processing Systems 26, 2013. ← (1) Statistics Decisions Theory における Springer Series in Statistics, SPRING ERVERLAG, New York, 1986. MR856411 ← 102, 160, 162 [34 ] E. L. L. Lehmann and G. Casella, Theory of Point Estimation, 2nd Ed., Springer Texts in Statistics, Springer-Verlag, New York, 1998. MR1639875 ← 160 I. VA.

Optim. 19 (2008), no. 4, 1574-1609, doi 10. In such cases, "optim" is either "optim" or not "optim", or either "optim" or not "optim", or not "optim", or not "optim", or not "optim". In this paper, "Complexity of Problems and Methods-Efficiency in Optimization", Wileyinterscience, John Wiley & amp; amp; amp; Sons, Inc, New York, 1983. MR702836 ← 102, 123, 157, 171 [42] A. Nemirovsky, Effective methods in convex programming, 1994. ← 171 [43] A. Nemirovsky, Lectures on modern convex optimisation, 2005. Georgia Institute of Technology. ← 140 [44] Yu. MR2142598 ← 124, 140, 171 [45] Y. Nesterov and A. Nemirovskii, Interior-point polynomial algorithms in convex programming, SIAM Studies in Applic Mathematics, vol. 13, Society in Industrial Applied Mathematics, 2005. 13, Society for Industrial and Applied Mathematics (SIAM) , Philadelphia, PA, 1994. MR1258086 ← 123 [46] J. Nocedal and S. J. Wright, Numerical Optimization, 2nd ed., Springer Series in Operations Research and Financial Engineering, Springer, NEW York, 2006. MR2244940 ← 101, 157 [47] B. T. Pole, Introduction to Optimization, a series of translations in mathematics and engineering, Optimization Software, Publications Division, New York, 1987. Translated from Russian; with a preface by Dimitry P. Bertsekas. MR1099605 ← 101, 140 [48] B. T. Paul, A. B.

[51] p. Shalev-Schwartz, Onami-Greeting: Theory, Algoriet and Applications, Doctor's ← Shair En S. Bounding En Transducing, J. Mach. Leren. Res. 15 (2014), 3401-3423. 3401-3423. Mr3277164 ← 102 [53] S. Shalev-Shwartz En T. Zhang, Stochastische Dubbele Coördinaat ascentieMethoden voor Geregularize Verliesminimalisatie, J. Mach. Leren. Res. 14 (2013), 567-599. MR303340 ← 140 [54] A. Shapiro, D. Dentcheva EN A. Ruszczynski, ´ Lectures on stochastic proogramming, MPS/SIAM Series ON Optimization, Vol. 9, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, Pa; Mathematical Programming Society (MPS), Philadelphia, PA, 2009. Modeling and theory. MR2562798 ← 102 [55] N. Z. Shor, Minimalisatiemethoden Voor Niet - Verschillende Functies, Springer Series in Computation Mathematics, Vol. 3, Springer-Werlag, Berlin, 1985. Translation from R Echter to K. K. Kivel and a. Rachinski. MR775136 ← 157 [56] n. Z. Shor, Investigated Optimization and Polynomial tasks, no n-mixed Oanem Publishers, Kluem, Kluem, Kluem. 1998. MR1620179 ← 153, 157 [57] R. Tibshirani, Regressie Krimp en Selectie Via de Lasso, J. Roy. Statist. SOC. Ser. B 58 (1996), nr. 1, 267-288. Mr1379242 ← 129 [58] a. B. Tsybakov, Introduction to no n-parametric Tsanging, Springer, 2009. ← 161, 171 [59] a. Wald, contribution to the theory of statisticswell evaluation and testing of hypotheses, a

10. 1090/PCMS/025/04 IAS/Park City Mathematics Series Volume 25, Pagina's 187-229 https: // doi. Org/10. 1090/PCMS/025/00832

Unreasonable method of matrix calculation PERE-GUNNAR MARTINSONON

Introduction 1. 1 Used area and purpose 1. 2 Main concept of randomization methods of low ranks 1, 3 Randomization methods excellent points 1. 4 pseudo Method of Test-Testing Personal Finance How to Single Passion 5. 1 Ermite matrix 5. 2 Binding matrix 6 How to have the complexity of O (MN log K K) for an optimistic queue 7 abstract equipment 7. 2 Limited probability of large deviation 8. Randomization scheme to increase accuracy 8. 3 Extended Collection Translation 9 Methods for No n-symmetric specified matrix 10 Randomized methods for calculating interpolation lag 10. 1. In office with factor decomposition 10. 2 3 ID: Right, lowercase letters. CUR-Expansion Calculation 11.

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© 2018 South American Mathematics Conversation

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

11. 2 A corresponding rank determination by updating the reconstructive matrix of both sides of the CUR output 12.

213 214 214 214 215 217 217 217 221 221 223 224 224 225

1. Introduction 1. The tasks and tasks of the administrator provided are a description of a series of randomized methods for effective calculations for the lo w-ranked approach of this matrix. In other texts, we want to determine the coefficient E and F in this matrix A-volume mx n.

Introduction 1. 1 Used area and purpose 1. 2 Main concept of randomization methods of low ranks 1, 3 Randomization methods excellent points 1. 4 pseudo Method of Test-Testing Personal Finance How to Single Passion 5. 1 Ermite matrix 5. 2 Binding matrix 6 How to have the complexity of O (MN log K K) for an optimistic queue 7 abstract equipment 7. 2 Limited probability of large deviation 8. Randomization scheme to increase accuracy 8. 3 Extended Collection Translation 9 Methods for No n-symmetric specified matrix 10 Randomized methods for calculating interpolation lag 10. 1. In office with factor decomposition 10. 2 3 ID: Right, lowercase letters. CUR-Expansion Calculation 11.

188 188 188 189 190 191 191 191 192 193 193 195 195 195 195 195 195 2002

© 2018 South American Mathematics Conversation

Randomization method of matrix calculation

11. 2 A corresponding rank determination by updating the reconstructive matrix of both sides of the CUR output 12.

213 214 214 214 215 217 217 217 221 221 223 224 224 225

1. Introduction 1. The tasks and tasks of the administrator provided are a description of a series of randomized methods for effective calculations for the lo w-ranked approach of this matrix. In other texts, we want to determine the coefficient E and F in this matrix A-volume mx n, in fact (1. 1. 1.) No n-row calculation method Per e-Gunnar Martinson

Introduction 1. 1 Used area and purpose 1. 2 Main concept of randomization methods of low ranks 1, 3 Randomization methods excellent points 1. 4 pseudo Method of Test-Testing Personal Finance How to Single Passion 5. 1 Ermite matrix 5. 2 Binding matrix 6 How to have the complexity of O (MN log K K) for an optimistic queue 7 abstract equipment 7. 2 Limited probability of large deviation 8. Randomization scheme to increase accuracy 8. 3 Extended Collection Translation 9 Methods for No n-symmetric specified matrix 10 Randomized methods for calculating interpolation lag 10. 1. In office with factor decomposition 10. 2 3 ID: Right, lowercase letters. CUR-Expansion Calculation 11.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

© 2018 South American Mathematics Conversation

Randomization method of matrix calculation

11. 2 A corresponding rank determination by updating the reconstructive matrix of both sides of the CUR output 12.

213 214 214 214 215 217 217 217 221 221 223 224 224 225

1. Introduction 1. The tasks and tasks of the administrator provided are a description of a series of randomized methods for effective calculations for the lo w-ranked approach of this matrix. In other texts, we want to determine the coefficient E and F in this matrix A-volume mx n, in fact (1. 1. 1.)

In one setting, rank K is set in advance, and in other tasks, it is performed to modify such a rank so that the approximation is restricted in the given rate in the queue and - -the part (this). In the section, only the Spectrornolm and Flobenius Norm are discussed). For the outer shape approximation (1. 1. 1), the queue A and F with the amount of K (m + n) are protected by the difference of the amount of MN, and A is A. It is useful for efficient calculation, interpretation of data, and almost everything else in the vector work of the matrix Z = ax (via y = fx and z = eY). The difficulty of approximating the low rank in similarity is considered to be the basis of data analysis and scientific calculation, and the structure of the test components (PCA) in computer statistics, the spectrum method for clustering hig h-magnification data, and the structure of the graph. It is widely applied to exploring, images and videos compression, modeling models in physiological modeling, and almost all other applications.

When doing low rankings, they are usually interested in numerical decomposition, which usually meets the additional restriction restrictions between the E and F coefficient. When A is a symmetrical matrix N × N, they are generally interested in seeking estimated attenuation (EVD), including images (1. 1. 2).

Here, column U forms a large orthogonal volume, and D is diagonal. We are usually interested in seeking the estimated number (SVD) of the specific values, including images, for the binding matrix A based on the volume of M × N (1. 1. 1. 3). < SPAN> In some settings, rank K is set in advance, and in other tasks, approximation is performed to modify such ranks so that it satisfies the given rates given by the matrix. (In this section, only the spectro norm and flobenius Norm). For the outer shape approximation (1. 1. 1), the queue A and F with the amount of K (m + n) are protected by the difference of the amount of MN, and A is A. It is useful for efficient calculation, interpretation of data, and almost everything else in the vector work of the matrix Z = ax (via y = fx and z = eY). The difficulty of approximating the low rank in similarity is considered to be the basis of data analysis and scientific calculation, and the structure of the test components (PCA) in computer statistics, the spectrum method for clustering hig h-magnification data, and the structure of the graph. It is widely applied to exploring, images and videos compression, modeling models in physiological modeling, and almost all other applications.

When doing low rankings, they are usually interested in numerical decomposition, which usually meets the additional restriction restrictions between the E and F coefficient. When A is a symmetrical matrix N × N, they are generally interested in seeking estimated attenuation (EVD), including images (1. 1. 2).

Here, column U forms a large orthogonal volume, and D is diagonal. We are usually interested in seeking the estimated number (SVD) of the specific values, including images, for the binding matrix A based on the volume of M × N (1. 1. 1. 3). In one setting, rank K is set in advance, and in other tasks, it is performed to modify such a rank so that the approximation is restricted in the given rate in the queue and - -the part (this). In the section, only the Spectrornolm and Flobenius Norm are discussed). For the outer shape approximation (1. 1. 1), the queue A and F with the amount of K (m + n) are protected by the difference of the amount of MN, and A is A. It is useful for efficient calculation, interpretation of data, and almost everything else in the vector work of the matrix Z = ax (via y = fx and z = eY). The difficulty of approximating the low rank in similarity is considered to be the basis of data analysis and scientific calculation, and the structure of the test components (PCA) in computer statistics, the spectrum method for clustering hig h-magnification data, and the structure of the graph. It is widely applied to exploring, images and videos compression, modeling models in physiological modeling, and almost all other applications.

When doing low rankings, they are usually interested in numerical decomposition, which usually meets the additional restriction restrictions between the E and F coefficient. When A is a symmetrical matrix N × N, they are generally interested in seeking estimated attenuation (EVD), including images (1. 1. 2).

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

U and V are regular orthodox, and D is diagonal. This chapter explains in detail about EVD and SVD. In addition, this factor decomposition is described as intermediate decomposition (ID) and CUR decomposition, which is very useful for interpretation of data and application to scientific calculations. Among them, we hope to modify the column (line) part of the column (line) from those that form a good foundation for the position of the column (line). However, in the last few sections, which aims to calculate the lo w-ranking factor decomposition, which is much smaller than the dimensions of the matrix M and N, the huge share of the leader is still randomized. The possibility of applied to accelerate the factor in the complete matrix factor, these are all of the SVDs required to deflate the QR with a turn to the absolute column, or to close the minimum square task. Type relaxation, etc. 1, 2. Explain the prototypes of unpretentious randomization methods to actively adopt the central ideas of the main ideas provided with low rank randomization methods. In other texts, we actually want a integer amount K 0. During this time, the pseudo button matrix A is deemed to be a queue N × m,

To any matrix A, a queue ∗ a † a = vk vk,

These are orthogonal projections to the rows of A and columns. If A is not infinite, it is a † = a-1.

3. The close-up of the offered matrix-The alignment task, which performs infinite decomposition two stages, is comfortable for two different "steps". This breakdown is explained for tasks that calculate estimated decomposition according to the specific value for embodiment. Conversely, the given matrix is the volume of Mx's and the number of K floors, and the coefficient U, D, V is determined. < SPAN> U and V are regular ort h-line, and D is diagonal. This chapter explains in detail about EVD and SVD. In addition, this factor decomposition is described as intermediate decomposition (ID) and CUR decomposition, which is very useful for interpretation of data and application to scientific calculations. Among them, we hope to modify the column (line) part of the column (line) from those that form a good foundation for the position of the column (line). However, in the last few sections, which aims to calculate the lo w-ranking factor decomposition, which is much smaller than the dimensions of the matrix M and N, the huge share of the leader is still randomized. The possibility of applied to accelerate the factor in the complete matrix factor, these are all of the SVDs required to deflate the QR with a turn to the absolute column, or to close the minimum square task. Type relaxation, etc. 1, 2. Explain the prototypes of unpretentious randomization methods to actively adopt the central ideas of the main ideas provided with low rank randomization methods. In other texts, we actually want a integer amount K 0. During this time, the pseudo button matrix A is deemed to be a queue N × m,

To any matrix A, a queue ∗ a † a = vk vk,

These are orthogonal projections to the rows of A and columns. If A is not infinite, it is a † = a-1.

3. The close-up of the offered matrix-The alignment task, which performs infinite decomposition two stages, is comfortable for two different "steps". This breakdown is explained for tasks that calculate estimated decomposition according to the specific value for embodiment. Conversely, the given matrix is the volume of Mx's and the number of K floors, and the coefficient U, D, V is determined. U and V are regular orthodox, and D is diagonal. This chapter explains in detail about EVD and SVD. In addition, this factor decomposition is described as intermediate decomposition (ID) and CUR decomposition, which is very useful for interpretation of data and application to scientific calculations. Among them, we hope to modify the column (line) part of the column (line) from those that form a good foundation for the position of the column (line). However, in the last few sections, which aims to calculate the lo w-ranking factor decomposition, which is much smaller than the dimensions of the matrix M and N, the huge share of the leader is still randomized. The possibility of applied to accelerate the factor in the complete matrix factor, these are all of the SVDs required to deflate the QR with a turn to the absolute column, or to close the minimum square task. Type relaxation, etc. 1, 2. Explain the prototypes of unpretentious randomization methods to actively adopt the central ideas of the main ideas provided with low rank randomization methods. In other texts, we actually want a integer amount K 0. During this time, the pseudo button matrix A is deemed to be a queue N × m,

To any matrix A, a queue ∗ a † a = vk vk,

These are orthogonal projections to the rows of A and columns. If A is not infinite, it is a † = a-1.

N $ N $ P+ J-p -j $ N $ P+ J-P -J $ N DKL P+J | P-J TV TV 2 J = 1

The coefficient of U and V is regular orthogonal, and D must be diagonal. (We hope that Rank K is actually popular in advance, but the method of dropping this guess is described in section 12). In accordance with 26], this task is divided into two calculation steps: period A-estimated spectra intelligence: ≈ QQ ∗ q。。。。。。。。。 The period B is a compilation of the given factor decomposition: a matrix Q is taken into account, calculated on line A, and the motorcycle U, D, and V are formed by applying a traditional deterministic method. In this period, for example, it can be implemented in the following appropriate steps: (1) K × N B = Q * A. ∗. (2) Calculate the SVD (small) matrix B that becomes b = udv ˆ (3) form a u = quu. When K (M, N) is drawn from K, all parts that are difficult to calculate takes time A. Remarks 3. The B stage is accurate in the mathematics of floating point, and as a result, all mistakes in the facto r-decomposition process appear in the A stage. B = QU q q ∗.

In other texts, if the moment Q meets the conditions of A-QQ * a ε, mechanically (3. 0.)

A-udv ∗ = a - qq ∗ a ε

When ε is not close to mechanical accuracy. Remarks 3. From the limit of (3. 0. 2) in the footprint, in fact, the diagonal component of D

It is regarded as an accurate approximation of a peculiar value A. However, the same limit as (3. 0. 2) does not distinguish the key of the conditional error of the peculiar value. In a general case, it does not distinguish between strong columns and such columns, and in fact, column U and V are considered a good approximation of a unique vector A.

Randomization method of matrix calculation

4. Stage A "Randomization Method s-Spectral Search Task In this section, explain the randomization method to complete the spectral task presented as" Stage A "in section 3. As a preparation for consideration," Ideal for consideration " Remember that the basal matte Q is a matrix UK for the spectrum of the provided queue A, and a similar vector A on the left-left of the intellectual K. < SPAN> The coefficient of U and V is a regular orthogonal, and D must be diagonal. (We hope that Rank K is actually popular in advance, but the method of dropping this guess is described in section 12). In accordance with 26], this task is divided into two calculation steps: period A-estimated spectra intelligence: ≈ QQ ∗ q。。。。。。。。。 The period B is a compilation of the given factor decomposition: a matrix Q is taken into account, calculated on line A, and the motorcycle U, D, and V are formed by applying a traditional deterministic method. In this period, for example, it can be implemented in the following appropriate steps: (1) K × N B = Q * A. ∗. (2) Calculate the SVD (small) matrix B that becomes b = udv ˆ (3) form a u = quu. When K (M, N) is drawn from K, all parts that are difficult to calculate takes time A. Remarks 3. The B stage is accurate in the mathematics of floating point, and as a result, all mistakes in the facto r-decomposition process appear in the A stage. B = QU q q ∗.

In other texts, if the moment Q meets the conditions of A-QQ * a ε, mechanically (3. 0.)

A-udv ∗ = a - qq ∗ a ε

When ε is not close to mechanical accuracy. Remarks 3. From the limit of (3. 0. 2) in the footprint, in fact, the diagonal component of D

It is regarded as an accurate approximation of a peculiar value A. However, the same limit as (3. 0. 2) does not distinguish the key of the conditional error of the peculiar value. In a general case, it does not distinguish between strong columns and such columns, and in fact, column U and V are considered a good approximation of a unique vector A.

Randomization method of matrix calculation

4. Stage A "Randomization Method s-Spectral Search Task In this section, explain the randomization method to complete the spectral task presented as" Stage A "in section 3. As a preparation for consideration," Ideal for consideration " Remember that the basal matte Q is a matrix UK for the spectrum of the provided queue A, and a similar vector A on the left-left of the intellectual K. The coefficient of U and V is regular orthogonal, and D must be diagonal. (We hope that Rank K is actually popular in advance, but the method of dropping this guess is described in section 12). In accordance with 26], this task is divided into two calculation steps: period A-estimated spectra intelligence: ≈ QQ ∗ q。。。。。。。。。 The period of B is a compilation of the given factor decomposition: the matrix Q is calculated on the line A, and a traditional deterministic method is applied to form a moment U, D, and V. In this period, for example, it can be implemented in the following appropriate steps: (1) K × N B = Q * A. ∗. (2) Calculate the SVD (small) matrix B that becomes b = udv ˆ (3) form a u = quu. When K (M, N) is drawn from K, all parts that are difficult to calculate takes time A. Remarks 3. The B stage is accurate in the mathematics of floating point, and as a result, all mistakes in the facto r-decomposition process appear in the A stage. B = QU q q ∗.

In other texts, if the moment Q meets the conditions of A-QQ * a ε, mechanically (3. 0.)

A-udv ∗ = a - qq ∗ a ε

When ε is not close to mechanical accuracy. Remarks 3. From the limit of (3. 0. 2) in the footprint, in fact, the diagonal component of D

Lecture on randomization numeric lectures Petros DRINEAS and MICHAEL V. MAHONI

Randomization method of matrix calculation

4. Stage A "Randomization Method s-Spectral Search Task In this section, explain the randomization method to complete the spectral task presented as" Stage A "in section 3. As a preparation for consideration," Ideal for consideration " Remember that the basal matte Q is a matrix UK for the spectrum of the provided queue A, and a similar vector A on the left-left of the intellectual K.

= [ Think about how to do it: Take a random vector K J = 1 from the gauss distribution, compare them with the range A vector YJ = AGJ, and use the resulting set K J = 1. For example, after the orthogonalized, a regular orthogonal base K J = 1 is obtained according to the gram-shMIDT. Special when a queue A has a accurate rank K, the vector k j = 1 can prove that it is linearly independent with 1 probability, and the obtained regular orthogonal base k J = 1 covers the range A accurately. The problem is that there is almost always a lot of no n-zero varieties outside the first K. The left special vector associated with these fixes "contaminates" the vector of the specimen YJ = AGJ, so the space covered with K J = 1 shifts, and is an ideal that is covered by K-K -sized vector A. It will not match the space. Fortunately, this can be solved. For example, if you take a k + 10 sample instead of a K, it can be seen that this process will give a basal as a basis with a probability close to 1. In summary, < Span> = | UK U ∗ k a || = σk+1 (a). Simplification to configure a set of a vector of K-bed in the range of matrix A. Consider the randomized method: Take a random vector K J = 1 from the Gaussian distribution, compare them with the range A vector Yj = AGJ, and as a result Use. For example, after the orthogonalized, a regular orthogonal base K J = 1 is obtained according to the gram-shMIDT. Special when a queue A has a accurate rank K, the vector k j = 1 can prove that it is linearly independent with 1 probability, and the obtained regular orthogonal base k J = 1 covers the range A accurately. The problem is that there is almost always a lot of no n-zero varieties outside the first K. The left special vector associated with these fixes "contaminates" the vector of the specimen YJ = AGJ, so the space covered with K J = 1 shifts, and is an ideal that is covered by K-K -sized vector A. It will not match the space. Fortunately, this can be solved. For example, if you take a k + 10 sample instead of a K, it can be seen that this process will give a basal as a basis with a probability close to 1. In summary = | A-uk u ∗ k a || = σk+1 (a). Here, the random random to configure a bed set to configure a set of k vectors for the range of matrix A. Think of the converted method: Take a random vector K J = 1 of K from Gaussian distribution, compare them with the range A vector Yj = AGJ, and use the resulting group K J = 1. For example, after the orthogonalized, a regular orthogonal base K J = 1 is obtained according to the gram-shMIDT. Special when a queue A has a accurate rank K, the vector k j = 1 can prove that it is linearly independent with 1 probability, and the obtained regular orthogonal base k J = 1 covers the range A accurately. The problem is that there is almost always a lot of no n-zero varieties outside the first K. The left special vector associated with these fixes "contaminates" the vector of the specimen YJ = AGJ, so the space covered with K J = 1 shifts, and is an ideal that is covered by K-K -sized vector A. It will not match the space. Fortunately, this can be solved. For example, if you take a k + 10 sample instead of a K, it can be seen that this process will give a basal as a basis with a probability close to 1. Summarized

"Inbox is theoretically close to optimal. Details are shown in section 7.

Algorithm: RSV D-Basic randomization SVD

Entrance: Starts A-Major M × N, target Lank K, excess sample parameter P (eg, p = 10). Output: Estimated rank (K + P) SV D-matrix A matrix U, D, V (U and V are deleted, D is diagonal, A ≈ UDV ∗.) Stage a: (1) n × (k (K + P)) Gaussandum matrix G. G. G = Randn (N, K + P) (2) Sample matrix y = a g. Y = a*g (3) Sample matrix Q = orth (y) column. [Q , ~] = QR (y, 0) phase b: (4) form (4) form (k + p) × n matrix b = q q ∗ q '*a ∗ ∗.... 立 立 立 立 立 立 立 立 立 5 : B = Udv [uHat, d, v] = svd (b, 'econ') ˆ (6) U = qua. U = Q*uHat

Algorithm 4. 0. 1. Basic randomized algorithm. If the factor is accurately required to rank K, the factor in step 5 can be cut off with koonstadigen. The San s-serif text below each line is a Matlab code for its implementation.

Remarks 4. 0. 2 (What is the number of basement vectors?) Our goal was to find a matrix Q including a regular accident of K, but the randomization process composed of 4. 0. 1-algorithm is discussed in this section. Readers will notice that they will guide a line with a column of K + P. Additional pecals are required to accurately represent the dominant left wing alpine vector a. b., which is obtained in phase a ". From a practical point of view, transfer several additional samples in the middle step. The cost is often not important at all.

5. The randomized algorithm described in the algorithm 4. 0. 1 is executed twice in the matrix A.

Randomization method of matrix calculation < SPAN> "Inbox is theoretically optimal. Details are shown in section 7.

Algorithm: RSV D-Basic randomization SVD

Algorithm: RSV D-Basic randomization SVD

Algorithm 4. 0. 1. Basic randomized algorithm. If the factor is accurately required to rank K, the factor in step 5 can be cut off with koonstadigen. The San s-serif text below each line is a Matlab code for its implementation.

Remarks 4. 0. 2 (What is the number of basement vectors?) Our goal was to find a matrix Q including a regular accident of K, but the randomization process composed of 4. 0. 1-algorithm is discussed in this section. Readers will notice that they will guide a line with a column of K + P. Additional pecals are required to accurately represent the dominant left wing alpine vector a. b., which is obtained in phase a ". From a practical point of view, transfer several additional samples in the middle step. The cost is often not important at all.

5. The randomized algorithm described in the algorithm 4. 0. 1 is executed twice in the matrix A.

Algorithm: RSV D-Basic randomization SVD

Algorithm: RSV D-Basic randomization SVD

Entrance: Starts A-Major M × N, target Lank K, excess sample parameter P (eg, p = 10). Output: Estimated rank (K + P) SV D-matrix A matrix U, D, V (U and V are deleted, D is diagonal, A ≈ UDV ∗.) Stage a: (1) n × (k (K + P)) Gaussandum matrix G. G. G = Randn (N, K + P) (2) Sample matrix y = a g. Y = a*g (3) Sample matrix Q = orth (y) column. [Q , ~] = QR (y, 0) phase b: (4) form (4) form (k + p) × n matrix b = q q ∗ q '*a ∗ ∗.... 立 立 立 立 立 立 立 立 立 5 : B = Udv [uHat, d, v] = svd (b, 'econ') ˆ (6) U = qua. U = Q*uHat

Algorithm 4. 0. 1. Basic randomized algorithm. If the factor is accurately required to rank K, the factor in step 5 can be cut off with koonstadigen. The San s-serif text below each line is a Matlab code for its implementation.

Algorithm: RSV D-Basic randomization SVD

5. The randomized algorithm described in the algorithm 4. 0. 1 is executed twice in the matrix A.

Randomization method of matrix calculation

In "Stage B", A, covered by the calculated basal vector, is projected into that place. It turns out that it is possible to change the method in this way, but in fact, I only go to a record. This is basically because the decomposition of the matrix can be determined, which is very large for saving. In the case of the Ermit matrix, the conversion of the method 4. 0. 1 is very easy, and is described in a measure 5. 1. Furthermore, in Section 5. 2, we will consider the case of a joint line. Remarks (accuracy loss). The single path method described in this section, in principle, distinguishes the accuracy of factor decomposition than those that may provide 4. 0. 1 law. For example, in a situation where either on e-path method or tw o-path method must be selected, the final method is generally better. Note (streaming algorithm). In fact, if we can use the queue only once, and if we do not count, there is a chance to submit all of the recording, the queue processing method is considered a streaming method. I have that. (In other sentences, this method is not allowed to establish a procedure to see the elements). How to explain in this section

U is a regular orthogonal matrix, and D is a diagonal matrix. (Note that EVD and SVD are essentially equivalent in the Ermit matrix, and that EVD is actually more naturally decomposed.) After implementing the period A with the parameter of the additional selection P for calculating the regular orthogonal matrix Q, forms a basal column close to the position of the column A: (1) Gaussland Dam matrix of volume N × (k + p) g. (2) Collection matrix Y = AG is formed. (3) The column Y is organized to form Q. During this time (5. 1. 2)

In Hermitov, (5. 1. 3) increases because the row is similar to the line.

Replacing (5. 1. 2) with (5. 1. 3) will actually find (unofficial!) (5. 1. 4). < SPAN> and "Stage B" projected A covered by the calculated basal vector. It turns out that it is possible to change the method in this way, but in fact, I only go to a record. This is basically because the decomposition of the matrix can be determined, which is very large for saving. In the case of the Ermit matrix, the conversion of the method 4. 0. 1 is very easy, and is described in a measure 5. 1. Furthermore, in Section 5. 2, we will consider the case of a joint line. Remarks (accuracy loss). The single path method described in this section, in principle, distinguishes the accuracy of factor decomposition than those that may provide 4. 0. 1 law. For example, in a situation where either on e-path method or tw o-path method must be selected, the final method is generally better. Note (streaming algorithm). In fact, if we can use the queue only once, and if we do not count, there is a chance to submit all of the recording, the queue processing method is considered a streaming method. I have that. (In other sentences, this method is not allowed to establish a procedure to see the elements). How to explain in this section

U is a regular orthogonal matrix, and D is a diagonal matrix. (Note that EVD and SVD are essentially equivalent in the Ermit matrix, and that EVD is actually more naturally decomposed.) After implementing the period A with the parameter of the additional selection P for calculating the regular orthogonal matrix Q, forms a basal column close to the position of the column A: (1) Gaussland Dam matrix of volume N × (k + p) g. (2) Collection matrix Y = AG is formed. (3) The column Y is organized to form Q. During this time (5. 1. 2)

Algorithm: RSV D-Basic randomization SVD

Replacing (5. 1. 2) with (5. 1. 3) will actually find (unofficial!) (5. 1. 4). In "Stage B", A, covered by the calculated basal vector, is projected into that place. It turns out that it is possible to change the method in this way, but in fact, I only go to a record. This is basically because the decomposition of the matrix can be determined, which is very large for saving. In the case of the Ermit matrix, the conversion of the method 4. 0. 1 is very easy, and is described in a measure 5. 1. Furthermore, in Section 5. 2, we will consider the case of a joint line. Remarks (accuracy loss). The single path method described in this section, in principle, distinguishes the accuracy of factor decomposition than those that may provide 4. 0. 1 law. For example, in a situation where either on e-path method or tw o-path method must be selected, the final method is generally better. Note (streaming algorithm). In fact, if we can use the queue only once, and if we do not count, there is a chance to submit all of the recording, the queue processing method is considered a streaming method. I have that. (In other sentences, this method is not allowed to establish a procedure to see the elements). How to explain in this section

U is a regular orthogonal matrix, and D is a diagonal matrix. (Note that EVD and SVD are essentially equivalent in the Ermit matrix, and that EVD is actually more naturally decomposed.) After implementing the period A with the parameter of the additional selection P for calculating the regular orthogonal matrix Q, forms a basal column close to the position of the column A: (1) Gaussland Dam matrix of volume N × (k + p) g. (2) Collection matrix Y = AG is formed. (3) The column Y is organized to form Q. During this time (5. 1. 2)

In Hermitov, (5. 1. 3) increases because the row is similar to the line.

Replacing (5. 1. 2) with (5. 1. 3) will actually find (unofficial!) (5. 1. 4).

Algorithm: RSV D-Basic randomization SVD

In fact, using AQQ * ≒ a (see (5. 1. 3)) to approximately approximation of the correct ratio B (5. 1. 6): Q * Aqq * g q Q QL.

By combining (5. 1. 6) and (5. 1. 7), and ignoring an approximation error, C is defined as a conclusion of a collective primary equation.

However, it is necessary to guarantee that C will be sent, and as a result, the system will be redeveloped almost twice. After gathering everything, we will soon get a method to explain in the method 5. 1. 9. Method: Randomization EVD in the Hermit matrix

The volume of the entity Elmeat matrix N × N, the motivated rank K, the unnecessary selection parameter P (eg, p = 10). Weekend: Appeared matrix U and D-ran k-K -EVD A matrix A (formation queue N × K, D is the diagonal matrix K × K, A ≈ UDU ∗). Period A: (1) N × (k + P) Gauss Random matrix G. (2) Form a group matrix Y = a G. (3) Q is a regular orthogonal matrix, intellectual left defect y. period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B.

(4) The conclusion of the minimum tw o-square C q * g = q * y, which is appropriately averaged, is C. ˆ ˆ ˆ ˆ. (5) C = Ud ˆ (6)

Algorithm: RSV D-Basic randomization SVD

If there is information in the randomization method < Span> C in the queue, there is no problem after calculation: Calculate the elementary particles, determine ˆ ˆ ∗, determine U = QU, seek the EVD of C, C. Get = Ud ˆ ˆ ˆ ∗ qla. A ≈ Qcq ∗ = qd, the problem here is the fact that (5. 1. 5) cannot be considered because it is looking for a single path method. For this reason, if (5. 1. 5) is applied to Q * G, C (Q * G) = Q * AQQ * G.

In fact, using AQQ * ≒ a (see (5. 1. 3)) to approximately approximation of the correct ratio B (5. 1. 6): Q * Aqq * g q Q QL.

By combining (5. 1. 6) and (5. 1. 7), and ignoring an approximation error, C is defined as a conclusion of a collective primary equation.

However, it is necessary to guarantee that C will be sent, and as a result, the system will be redeveloped almost twice. After gathering everything, we will soon get a method to explain in the method 5. 1. 9. Method: Randomization EVD in the Hermit matrix

The volume of the entity Elmeat matrix N × N, the motivated rank K, the unnecessary selection parameter P (eg, p = 10). Weekend: Appeared matrix U and D-ran k-K -EVD A matrix A (formation queue N × K, D is the diagonal matrix K × K, A ≈ UDU ∗). Period A: (1) N × (k + P) Gauss Random matrix G. (2) Form a group matrix Y = a G. (3) Q is a regular orthogonal matrix, intellectual left defect y. period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B.

(4) The conclusion of the minimum tw o-square C q * g = q * y, which is appropriately averaged, is C. ˆ ˆ ˆ ˆ. (5) C = Ud ˆ (6)

Algorithm 5. Basic single resistance randomization method gives priority to the Hermit matrix.

Algorithm: RSV D-Basic randomization SVD

In fact, using AQQ * ≒ a (see (5. 1. 3)) to approximately approximation of the correct ratio B (5. 1. 6): Q * Aqq * g q Q QL.

By combining (5. 1. 6) and (5. 1. 7), and ignoring an approximation error, C is defined as a conclusion of a collective primary equation.

However, it is necessary to guarantee that C will be sent, and as a result, the system will be redeveloped almost twice. After gathering everything, we will soon get a method to explain in the method 5. 1. 9. Method: Randomization EVD in the Hermit matrix

The volume of the entity Elmeat matrix N × N, the motivated rank K, the unnecessary selection parameter P (eg, p = 10). Weekend: Appeared matrix U and D-ran k-K -EVD A matrix A (formation queue N × K, D is the diagonal matrix K × K, A ≈ UDU ∗). Period A: (1) N × (k + P) Gauss Random matrix G. (2) Form a group matrix Y = a G. (3) Q is a regular orthogonal matrix, intellectual left defect y. period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B: period B.

(4) The conclusion of the minimum tw o-square C q * g = q * y, which is appropriately averaged, is C. ˆ ˆ ˆ ˆ. (5) C = Ud ˆ (6)

Algorithm 5. Basic single resistance randomization method gives priority to the Hermit matrix.

Randomization method of matrix calculation

(1) A similar monitoring of the equation (5. 1. 4) is more common than (5. 1. 2) in principle. (2) However, the matrix Q * G is an inverted matrix and includes a fairly bad decision. Remarks 5. 1. 10 (additional over growth). If the memory allows, it is possible to introduce P = K. Then, at the end of the selection step, use the Q to be the main K lef t-win g-specific vector Y (form a complete SVD Y and calculate them to discard the latest PP components). During this time, C has the value of K × K, and the formula that defines C includes the following image.

(5. 1. 11) K × k k × k × In effect, the necessary information (queue K × k, not ×) has decreased, and more information has to be determined. 。 5. 2. The joint matrix will further see the volume M × N in the joint matrix. (1) Build two Gaussian random matrix: GC volume Nx (k + P) and GR volume mx (k + p). (3) Two base matrix QC = orth (YC) and QR = ORTH (JR). Here, a small imaging matrix is defined (5. 2. 1).

Similarly, two proportions that define C are derived (5. 1. 6). First, take the G * R QC on (5. 2. 1) and obtain (5. 2. 2).< i >G * R QC C = G * R QC Q QR ≈ G * R AQR = JR * QR.

Algorithm: RSV D-Basic randomization SVD

Here, C is defined as a conclusion of a smaller secondary equation.

Then, C Q ∗ R GC = Q ∗ C YC. When the GR QC C = Yr ∗ QR is r e-emitted, the system is reduced by about twice and is ranked even more useful by more cruel prescriptions. Method 5. 2. 4 is a summary of the single path method for the joint matrix. < SPAN> (5. 1. 4) approximation monitoring of equation (1) (5. 1. 4) is more common than (5. 1. 2). (2) However, the matrix Q * G is an inverted matrix and includes a fairly bad decision. Remarks 5. 1. 10 (additional over growth). If the memory allows, it is possible to introduce P = K. Then, at the end of the selection step, use the Q to be the main K lef t-win g-specific vector Y (form a complete SVD Y and calculate them to discard the latest PP components). During this time, C has the value of K × K, and the formula that defines C includes the following image.

(5. 1. 11) K × k k × k × In effect, the necessary information (queue K × k, not ×) has decreased, and more information has to be determined. 。 5. 2. The joint matrix will further see the volume M × N in the joint matrix. (1) Build two Gaussian random matrix: GC volume Nx (k + P) and GR volume mx (k + p). (3) Two base matrix QC = orth (YC) and QR = ORTH (JR). Here, a small imaging matrix is defined (5. 2. 1).

Similarly, two proportions that define C are derived (5. 1. 6). First, take the G * R QC on (5. 2. 1) and obtain (5. 2. 2).

G * R QC C = G * R QC Q QR ≈ G * R AQR = JR * QR.

Next, when the Q * R GC is applied to the right side of (5. 2. 1), (5. 2. 3) is obtained.< i >Here, C is defined as a conclusion of a smaller secondary equation.

Algorithm: RSV D-Basic randomization SVD

(5. 1. 11) K × k k × k × In effect, the necessary information (queue K × k, not ×) has decreased, and more information has to be determined. 。 5. 2. The joint matrix will further see the volume M × N in the joint matrix. (1) Build two Gaussian random matrix: GC volume Nx (k + P) and GR volume mx (k + p). (3) Two base matrix QC = orth (YC) and QR = ORTH (JR). Here, a small imaging matrix is defined (5. 2. 1).

Similarly, two proportions that define C are derived (5. 1. 6). First, take the G * R QC on (5. 2. 1) and obtain (5. 2. 2).

G * R QC C = G * R QC Q QR ≈ G * R AQR = JR * QR.

Algorithm: RSV D-Basic randomization SVD

Here, C is defined as a conclusion of a smaller secondary equation.

Then, C Q ∗ R GC = Q ∗ C YC. When the GR QC C = Yr ∗ QR is r e-emitted, the system is reduced by about twice and is ranked even more useful by more cruel prescriptions. Method 5. 2. 4 is a summary of the single path method for the joint matrix.

Algorithm: RSV D-Basic randomization SVD

Algorithm 5. Basic one randomated algorithm suitable for general matrix.

6. The algorithm of the randomation SVD (RSVD) of the complexity of the general secret string indicated in the algorithm 4. 0. 1 is a high-speed method for calculating the matrix-vector work x → axe. It is very effective if you can use it. If A is a total M × N matrix, it can be easily set in the form of a set of materials, and the calculation cost of the matrix-image Y = AG is O (MNK) in the step of algorithm 4. 0. 1. RSVD is often faster than classical techniques because it can optimize the multiplication of a queue-statue, but is not asymptotic. (1) Ω is freely structured, Aω can be calculated by O (mn (mn log (k)))), (2) Ω is freely random, and Aω columns are accurate the range of A. To cover. For example, the good choice of random matrix ω is (6. 0. 1).

Here D is a diagonal matrix, the diagonal note is a complex number of 1 module extracted from a uniform distribution above the complex plane, and F is F (P, Q), F (P, Q). = n-1/2 E-2π ( p-1) ( Q-1)/n,

Random method of matrix calculation

Algorithm: RSV D-Basic randomization SVD

7. Supply of performance This section simply summarizes some of the results of the output error of the basic RSVD method in the method 4. 0. 1. In fact, the coefficient U, D, V relies on the selection of random matrix G. As a result, the significant significance of failure is clearly restricted at the early stage, and there is also a possibility that it will deviate greatly from this expected value. The fundamental cause is that "pos t-processing" is considered in milestone B. In other words, we can limit ourselves to the restrictions on A-QQ ∗ A. Remark 7. 0. 1. The abstract research of mistakes caused by the use of randomization methods in the lin e-type algebra depends greatly on the concept of random matrix, theoretical computational machine science, traditional numerical line algebra, and almost all other fields. It is considered to be an intensive research field. Our job is simply to present several typical results, and there is no proof or details.

1. The most important result of expectations is [26] axios 10. 6. Expectation errors The most important consequence on the normal observed errors is [26] axiosity 10. 6, which is as follows: min (m, n).

Theorem 7. A shall be M × N matrix with a specific value j = 1. K is the motivation and rank p, and p 2 and p 2 and k + p min (m, n) are the same recampling parameters. G is a Gaus Randam matrix of Nx (K + P) and gives Q = Orth (AG). During this time, the average error measured by Flobenius Norm will meet the conditions of ⎞ 1/2 1/2 min (m, n).

K ⎝ σ ⎝ ⎝ ⎝, (7. 1. 2) E a-QQ ∗ AFRO 1 + p-1 j = k + 1

Here E represents the mathematical expected value of G. The consistent fruit of the measured values that are generally accepted will be as follows: ⎞1/2 ⎛ min (m, n)

K E K+P ⎝ (7. 1. 3) E a-QQ ∗ A 1+σk+1+σ2j ⌊. Easy summarizes some of the results of the output error of the RSVD method. In fact, the coefficient U, D, V depends on the selection of random matrix G as well as A. As a result, the significant significance of failure is clearly restricted at the early stage, and there is also a possibility that it will deviate greatly from this expected value. The fundamental cause is that "pos t-processing" is considered in milestone B. In other words, we can limit ourselves to the restrictions on A-QQ ∗ A. Remark 7. 0. 1. The abstract research of mistakes caused by the use of randomization methods in the lin e-type algebra depends greatly on the concept of random matrix, theoretical computational machine science, traditional numerical line algebra, and almost all other fields. It is considered to be an intensive research field. Our job is simply to present several typical results, and there is no proof or details.

1. The most important result of expectations is [26] axios 10. 6. Expectation errors The most important consequence on the normal observed errors is [26] axiosity 10. 6, which is as follows: min (m, n).

Theorem 7. A shall be M × N matrix with a specific value j = 1. K is the motivation and rank p, and p 2 and p 2 and k + p min (m, n) are the same recampling parameters. G is a Gaus Randam matrix of Nx (K + P) and gives Q = Orth (AG). During this time, the average error measured by Flobenius Norm will meet the conditions of ⎞ 1/2 1/2 min (m, n).

K ⎝ σ ⎝ ⎝ ⎝, (7. 1. 2) E a-QQ ∗ AFRO 1 + p-1 j = k + 1

Algorithm: RSV D-Basic randomization SVD

K E K+P ⎝ (7. 1. 3) E a-QQ ∗ A 1+σk+1+σ2j ⌊. p-1pj = k+17. A brief summary of some of the results of the output error. In fact, the coefficient U, D, V relies on the selection of random matrix G. As a result, the significant significance of failure is clearly restricted at the early stage, and there is also a possibility that it will deviate greatly from this expected value. The fundamental cause is that "pos t-processing" is considered in milestone B. In other words, we can limit ourselves to the restrictions on A-QQ ∗ A. Remark 7. 0. 1. The abstract research of mistakes caused by the use of randomization methods in the lin e-type algebra depends greatly on the concept of random matrix, theoretical computational machine science, traditional numerical line algebra, and almost all other fields. It is considered to be an intensive research field. Our job is simply to present several typical results, and there is no proof or details.

1. The most important result of expectations is [26] axios 10. 6. Expectation errors The most important consequence on the normal observed errors is [26] axiosity 10. 6, which is as follows: min (m, n).

Theorem 7. A shall be M × N matrix with a specific value j = 1. K is the motivation and rank p, and p 2 and p 2 and k + p min (m, n) are the same recampling parameters. G is a Gaus Randam matrix of Nx (K + P) and gives Q = Orth (AG). During this time, the average error measured by Flobenius Norm will meet the conditions of ⎞ 1/2 1/2 min (m, n).

K ⎝ σ ⎝ ⎝ ⎝, (7. 1. 2) E a-QQ ∗ AFRO 1 + p-1 j = k + 1

Here E represents the mathematical expected value of G. The consistent fruit of the measured values that are generally accepted will be as follows: ⎞1/2 ⎛ min (m, n)

K E K+P ⎝ (7. 1. 3) E a-QQ ∗ a 1+σk+1+σ2j ⌊. p-1pj = k+1

When the mistakes are measured normally with Flobenius, the axiac 7. 1. 1 is quite interesting. In the case of our normal opinion p = 10, we are on the host of 1 + k / 9 of the small error (actually ECCART A-an g-ang minus (M, n) 2 / 2σ's axiom = K + 1 Image should be the bottom of Remes of rank-K). Ove r-bolt in the selection and p = k + 1 is at a distance of 2 from a theoretical level with a small error. The situation is not the most tempted when mistakes are measured by spectral norm. The first paragraph of the restricted (7. 1. 3) is absolutely applicable, but for example, how to connect the minimum monitoring with Flobenius Norm, for example, is not successful. Especially if there are many M and N, it may be more likely. The extent of the (7. 1. 3) monitoring is determined by how much min (m, n) disappears. If they are rapidly declining, the surveillance of the measured values that are universally recognized in the spectrum of "Tail" specific, and the universally recognized advantages of FROBENIUS are the same. Yes, RSVD works perfectly. If it disappears slowly, the RSVD works completely.

Random method of matrix calculation

At the moment, β is not close to 1, but in this case, we are trying to make a small dedicated value contribution. Case 2-No attenuation: For example, σj = σk+1 for, for example, the fact that the peculiar average tail does not diverge, for example, J & amp; amp; gt; k. < SPAN> Measuring mistakes with Flobenius is quite interesting. In the case of our normal opinion p = 10, we are on the host of 1 + k / 9 of the small error (actually ECCART A-an g-ang minus (M, n) 2 / 2σ's axiom = K + 1 Image should be the bottom of Remes of rank-K). Ove r-bolt in the selection and p = k + 1 is at a distance of 2 from a theoretical level with a small error. The situation is not the most tempted when mistakes are measured by spectral norm. The first paragraph of the restricted (7. 1. 3) is absolutely applicable, but for example, how to connect the minimum monitoring with Flobenius Norm, for example, is not successful. Especially if there are many M and N, it may be more likely. The extent of the (7. 1. 3) monitoring is determined by how much min (m, n) disappears. If they are rapidly declining, the surveillance of the measured values that are universally recognized in the spectrum of "Tail" specific, and the universally recognized advantages of FROBENIUS are the same. Yes, RSVD works perfectly. If it disappears slowly, the RSVD works completely.

Random method of matrix calculation

At the moment, β is not close to 1, but in this case, we are trying to make a small dedicated value contribution. Case 2-No attenuation: For example, σj = σk+1 for, for example, the fact that the peculiar average tail does not diverge, for example, J & amp; amp; gt; k. When the mistakes are measured normally with Flobenius, the axiac 7. 1. 1 is quite interesting. In the case of our normal opinion p = 10, we are on the host of 1 + k / 9 of the small error (actually ECCART A-an g-ang minus (M, n) 2 / 2σ's axiom = K + 1 Image should be the bottom of Remes of rank-K). Ove r-bolt in the selection and p = k + 1 is at a distance of 2 from a theoretical level with a small error. The situation is not the most tempted when mistakes are measured by spectral norm. The first paragraph of the restricted (7. 1. 3) is absolutely applicable, but for example, how to connect the minimum monitoring with Flobenius Norm, for example, is not successful. Especially if there are many M and N, it may be more likely. The extent of the (7. 1. 3) monitoring is determined by how much min (m, n) disappears. If they are rapidly declining, the surveillance of the measured values that are universally recognized in the spectrum of "Tail" specific, and the universally recognized advantages of FROBENIUS are the same. Yes, RSVD works perfectly. If it disappears slowly, the RSVD works completely.

Algorithm: RSV D-Basic randomization SVD

At the moment, β is not close to 1, but in this case, we are trying to make a small specific value contribution. Case 2-No attenuation: For example, σj = σk+1 for, for example, the fact that the peculiar average tail does not diverge, for example, J & amp; amp; gt; k.

This is the worst scenario, which wants to make N and M quite huge, so it brings catastrophic quas i-appropriate. Fortunately, by adjusting the RSVD in this way, it is possible to make the most of the spectrum and, for example, in the Flobenius Norm, to make the mistakes obtained as a result. As a compensation, the calculation cost increases unreasonably. See Section 8 and [26, Para 4. 5]. 7. 2. The big difference from the average average to the possibility of a large deviation can only be only for recruitment parameters P (surprisingly), and can justify the decrease very rapidly. is. For example, if P is 4, √ 1/2 8 k + p * σ2j, (7. 2. 1) || a-QQ a || 1 + 17 1 + k/p σK + 1 + j & amp; amp; See [26, 10. 9], which can justify that the possibility of the problem does not exceed 3 E with gt; k p+ 1-p.

8. Randomization design to improve accuracy 8. 1. The main ideas are strong repetitions. In section 7, the basic randomization scheme (for example, the method 4. 0. 1) is clearly specified for a queue where the specific meaning decreases rapidly, but it is not optimal. I realized that there was a tendency to get results. Don't forget to calculate the approximation of min (m, n). In fact, this doctrine indicates that the error measured by spectro norm is limited by the moment to M × N matrix A with a specific value j = 1.

2/2. When the specific value is attenuated slowly, the given size is much more j & amp; amp; gt; k σj larger than the theoretical level of the minimum approximation error (this is equal to σK+1). There is. In fact, the purpose of the randomization selection is to build a set of regular orthogonal vector J = 1, which covers where the dominant left special vector K J = 1 covers the dominant left special vector K J = 1. Let's remember.

Algorithm: RSV D-Basic randomization SVD

In other sentences, A (Q) contains the same left special vector as A, and its singularity is & amp; amp; gt; j. Unique average, including when the peculiar average A is attenuated slowly

Q y = aa * ag. Hereafter, column Y is positive and Q = orth (y). The acquired schema is shown in 8. 1. 1. Method: Precision improvement Random SVD entity: matrix A volume M × N, motivation rank K, punk tization paragram P (eg, p = 10), small unit number Q indicating severe repetitive steps. Weekend: Appearance Rank (K + P) SVD A matrix U, D, V (that is, U and V are orthogonal, D is diagonal. (1) g = Randn (n, k + p); (2) y = Ag; (3) for j = 1: q (4) z = a ∗ y; (5) y = az; (6) end for (7) q = orth (y); (8) b = q ∗ a; ˆ d, v] = svd (b, 'econ'); (9) [u, ˆ (10) u = quu;

Algorithm 8. 1. 1. Randomization SVD accuracy improvement of accuracy is required, and factoral decomposition of K is required, step 9 factor decomposition may be cut off by the main member K. Note 8. 1. 2. Method 8. The scheme described in 1. 1 may reduce accuracy due to round errors. This contradiction is, in fact, when Q increases, all columns of specimens

Q Steering y = aa ∗ AG Eagers provides something closer to the dominant left vector. In fact, this means that all information about the specific value and the specific vector related to the specific value and the minimum specific value is related to errors. Boldly, if σj 1/(2Q+1) mach, σ1, here is the mechanical accuracy, all information related to J-m is a specific value, further stalling (section 3. 2). V [34]). As shown in, this contradiction may be solved by the orthogonal method of columns between all repetitions.

Randomization method of matrix calculation

Algorithm: RSV D-Basic randomization SVD

Algorithm: Randomized SVD accuracy

(1) (2) (3) (5) (6) (7) (9)

(Orthonomination) g = Randn (n, k + p); q = orth (AG); for j = 1: Q w = orth (a ∗ q); q = orth (aw); end for b = q ∗;; ˆ d, v] = svd (b, 'econ'); [u, ˆ u = quu;

Algorithm: RSV D-Basic randomization SVD

8. 2. Theoretical results Algorithm 8. 1. 1 is given in [26, section 10. 4]. In particular, the main theorem is the following: Theorem 8. Let A be M×N, P 2-matrix-overkletting parameter, and Q a small integer. Take a Gaussian matrix G of size N × (k + p), let y = (aa ∗) q ag, and let q be an orthonormal matrix of size m × (k + p).

E a - qq ∗ a ⎡1/2 ⎤(2q+ 1) √ min (m, n) (8. 2. 2) k e k+ p ⎥ 2 (2q+ 1) ⎥ 2+ σ2q+ 1+ σj. ⎦ k+1 p-1 p j = k+1

The restrictions in (8. 2. 2) are somewhat opaque. Consider the simplification of the worst case where σk+1 = σk+2 = - - - = σmin (m, n) since there is no singular value decay behind the cutoff point. Then (8. 2. 2) simplifies to 1/(2q+1)√.

E K+P K∗+ - min - K σk+1. e a - qq a 1 + p-1 p

That is, as Q increases, the power supply circuit exponentially increases the factor multiplied by σk+1 per unit. This factor is the expected degree of "non-optimality". 8. 3. The extended sample matrix method presented in subsection 8. 1 is somewhat wasteful since it does not directly use all the calculated sample vectors. To achieve even greater accuracy, we can form an "extended" sample matrix for the symmetric matrix A and a small positive integer Q.

Y = ag, a2 g ,. . . Note that this new sample matrix Y has Q columns. Then we act as before: (8. 3. 1)

ˆ d, v] = svd (b, 'econ'), [u,

Algorithm: RSV D-Basic randomization SVD

9. See NADOUX methods for the symmetrical specific matrix [9] and reference. To sketch this idea from Section 5. 1, let's actually sketch the Ermitov case A (of course this is any PSD matrix).

(9. 0. 1) a ≈ ≈ q q q q q q q ls. As a different point, “NYUSTRO M-schema” is based on the rank approximation.

-1 (AQ) ∗. (9. 0. 2) A≈ (AQ) Q ∗ In order to ensure the resistance and performance of AQ calculation, we usually rewrite (9. 0. 2) as A≈ff ∗. 。

-1/2. F = (aq) q ∗。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。 For example, the factor of cholesterol factor b2 = c * c support can be determined.

Any method of queue calculation

Algorithm: RSV D-Basic randomization SVD

(9. 0. 1) a ≈ ≈ q q q q q q q ls. As a different point, “NYUSTRO M-schema” is based on the rank approximation.

-1 (AQ) ∗. (9. 0. 2) A≈ (AQ) Q ∗ In order to ensure the resistance and performance of AQ calculation, we usually rewrite (9. 0. 2) as A≈ff ∗. 。

-1/2. F = (aq) q ∗。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。 For example, the factor of cholesterol factor b2 = c * c support can be determined.

Any method of queue calculation

Executes a triangular inference. Lo w-ranking factor decomposition (9. 0. 2) may be converted into normal decomposition by support of section 3 techniques. Let's compare the price of this method and the normal method when the formula (9. 0. 1) is used. In both cases, the K + P-vector set must be used twice (when calculating AG first and then calculating AQ). However, NADU method generally distinguishes clearer results. Unofficially, the fundamental cause is that the use of PSD Cedement A can control one step of the "freedom" time of the restoration. See [19, 43] for the official analysis of the NYUSTRA method price and accuracy. Method: NADUSTE's no n-negotent queue of N × n, the no n-negative queue of N × N, motivation rank K, and parameter P in the peregoquing A≈u λu ∗ (U is regular orthogonal, λ is no n-negative. It must be the procedure of the diagonal). (1) Take Gauss Random matrix G = Randn (n, k + p). 9. See NADOUX methods for the symmetrical specific matrix [9] and reference. To sketch this idea from Section 5. 1, let's actually sketch the Ermitov case A (of course this is any PSD matrix).<(sτ )τ∈X : sσ = F(ρσ)(sρ ) ∀ ρ σ>⊂ τ

(9. 0. 1) a ≈ ≈ q q q q q q q ls. As a different point, “NYUSTRO M-schema” is based on the rank approximation.

-1 (AQ) ∗. (9. 0. 2) A≈ (AQ) Q ∗ In order to ensure the resistance and performance of AQ calculation, we usually rewrite (9. 0. 2) as A≈ff ∗. 。

-1/2. F = (aq) q ∗。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。。 For example, the factor of cholesterol factor b2 = c * c support can be determined.

Any method of queue calculation

Algorithm: RSV D-Basic randomization SVD

(2) Collection matrix Y = AG is formed. (3) The row of the matrix is Ortonz and obtain the basal matte Q = orth (y). (4) matrix B1 = AQ and B2 = Q * B1. (5) Complete the riding floor of Cholesterol B2 = C * C. (6) Form a F = B1 C-1 with the support of the triangular word. (7) Calculate the specific decomposition of cholesterol coefficient [u, σ, ~] = SVD (f, 'econ'). (8) λ = σ2.

Algorithm 9. 0. 3. NADO method for low rank approximation of sel f-consolidated matrix with no n-negative personal values. This is the same price as the 4. 0. 1-RSV D-bass of 4. 0. 1-RSV D-bass of 4. 0. 1 to the K + P-a column. However, a symmetric A is used to increase accuracy.

10. Random calculation method of interpolation attenuation 10. Save the structure with the volume of Motoral decomposition M × n, the rank k ε k = k+1 matrix is a vector y ∈ Ran (ak-1) q = y/y. ; B = q * AK-1; [QK-1 Q]; qk = BK-1 BK =; B understands.

(9) ak = a k-1 - qb; (10) End for

An algorithm 12. 2. 1. This is an unfamiliar method that sets the lo w-order approximation of the volume m x n of this matrix A in this sense ε. Conversely, this method determines the regular orthogonal matrix QK with a single volume k and the volume k × n, the matrix of the volume k × n, so in fact a-qk bk ε ε. You just have to understand that A = QJ BJ + AJ is in step J.

Algorithm: RSV D-Basic randomization SVD

The 12. 1 algorithm is considered to be a generalization of traditional Gramma-ShmidT procedures. The recognition source of this method is assigned a similarity A = QJ BJ + AJ,

The calculation efficiency and accuracy of this method vary greatly depending on the selection strategy of the vector Y on the row. Let's look at the three options: For example, let's say, for example, select a larger preserved column to instruct (4) rows and give Y a larger column than the AK-1 residual matrix. (4)

Algorithm: RSV D-Basic randomization SVD

Algorithm 9. 0. 3. NADO method for low rank approximation of sel f-consolidated matrix with no n-negative personal values. This is the same price as the 4. 0. 1-RSV D-bass of 4. 0. 1-RSV D-bass of 4. 0. 1 to the K + P-a column. However, a symmetric A is used to increase accuracy.

10. Random calculation method of interpolation attenuation 10. Save the structure with the volume of Motoral decomposition M × n, the rank k ε k = k+1 matrix is a vector y ∈ Ran (ak-1) q = y/y. ; B = q * AK-1; [QK-1 Q]; qk = BK-1 BK =; B understands.

(9) ak = a k-1 - qb; (10) End for

Algorithm: RSV D-Basic randomization SVD

Randomization method of matrix calculation

The 12. 1 algorithm is considered to be a generalization of traditional Gramma-ShmidT procedures. The recognition source of this method is assigned a similarity A = QJ BJ + AJ,

The calculation efficiency and accuracy of this method vary greatly depending on the selection strategy of the vector Y on the row. Let's look at the three options: For example, let's say, for example, select a larger preserved column to instruct (4) rows and give Y a larger column than the AK-1 residual matrix. (4)

JK = Argmax, and during this time y = ak-1 (:, jk). (2) Collection matrix Y = AG is formed. (3) The row of the matrix is Ortonz and obtain the basal matte Q = orth (y). (4) matrix B1 = AQ and B2 = Q * B1. (5) Complete the riding floor of Cholesterol B2 = C * C. (6) Form a F = B1 C-1 with the support of the triangular word. (7) Calculate the specific decomposition of cholesterol coefficient [u, σ, ~] = SVD (f, 'econ'). (8) λ = σ2.

Algorithm 9. 0. 3. NADO method for low rank approximation of sel f-consolidated matrix with no n-negative personal values. This is the same price as the 4. 0. 1-RSV D-bass of 4. 0. 1-RSV D-bass of 4. 0. 1 to the K + P-a column. However, a symmetric A is used to increase accuracy.

10. Random calculation method of interpolation attenuation 10. Save the structure with the volume of Motoral decomposition M × n, the rank k ε k = k+1 matrix is a vector y ∈ Ran (ak-1) q = y/y. ; B = q * AK-1; [QK-1 Q]; qk = BK-1 BK =; B understands.

(9) ak = a k-1 - qb; (10) End for

An algorithm 12. 2. 1. This is an insatiable method that sets the lo w-order approximation of this matrix A with accuracy ε in this sense. Conversely, this method determines the regular orthogonal matrix QK with a single volume k and the volume k × n, the matrix of the volume k × n, so in fact a-qk bk ε ε. You just have to understand that A = QJ BJ + AJ is in step J.

Randomization method of matrix calculation

The 12. 1 algorithm is considered to be a generalization of traditional Gramma-ShmidT procedures. The recognition source of this method is assigned a similarity A = QJ BJ + AJ,

The calculation efficiency and accuracy of this method vary greatly depending on the selection strategy of the vector Y on the line (4). Let's look at the three options: For example, select a larger preserved column to instruct (4) rows and give Y to a row larger than the AK-1 residual matrix. (4)

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