Linear Algebra and Optimization for Machine Learning A Textbook
This textbook introduces linear algebra and optimization in the context of machine learning. Examples and exercises are provided throughout the book. A solution manual for the exercises at the end of each chapter is available to teaching instructors. This
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Linear Algebra and Optimization for Machine Learning A Textbook
Linear Algebra and Optimization for Machine Learning
Charu C. Aggarwal
Linear Algebra and Optimization for Machine Learning A Textbook
Charu C. Aggarwal Distinguished Research Staff Member IBM T.J. Watson Research Center Yorktown Heights, NY, USA
ISBN 978-3-030-40343-0 ISBN 978-3-030-40344-7 (eBook) https://doi.org/10.1007/978-3-030-40344-7 © Springer Nature Switzerland AG 2020 This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG. The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
To my wife Lata, my daughter Sayani, and all my mathematics teachers
Contents
1 Linear Algebra and Optimization: An Introduction 1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Scalars, Vectors, and Matrices . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Basic Operations with Scalars and Vectors . . . . . . . . . . . 1.2.2 Basic Operations with Vectors and Matrices . . . . . . . . . . 1.2.3 Special Classes of Matrices . . . . . . . . . . . . . . . . . . . . 1.2.4 Matrix Powers, Polynomials, and the Inverse . . . . . . . . . 1.2.5 The Matrix Inversion Lemma: Inverting the Sum of Matrices 1.2.6 Frobenius Norm, Trace, and Energy . . . . . . . . . . . . . . 1.3 Matrix Multiplication as a Decomposable Operator . . . . . . . . . . 1.3.1 Matrix Multiplication as Decomposable Row and Column Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Matrix Multiplication as Decomposable Geometric Operators 1.4 Basic Problems in Machine Learning . . . . . . . . . . . . . . . . . . 1.4.1 Matrix Factorization . . . . . . . . . . . . . . . . . . . . . . . 1.4.2 Clustering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.3 Classification and
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