| Description |
1 online resource (vi, 402 pages) : illustrations |
| Contents |
Cover -- Half-title page -- Title page -- Copyright page -- Contents -- Preface -- 1 Introduction -- 1.1 Motivation: The Pitfalls of Large-Dimensional Statistics -- 1.1.1 The Big Data Era: When n Is No Longer Much Larger than p -- 1.1.2 Sample Covariance Matrices in the Large n,p Regime -- 1.1.3 Kernel Matrices of Large-Dimensional Data -- The Nontrivial Classification Regime -- Asymptotic Loss of Pairwise Distance Discrimination -- Explaining Kernel Methods with Random Matrix Theory -- Do Real Data Follow Small- or Large-Dimensional Intuitions? -- 1.1.4 Summarizing |
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1.2 Random Matrix Theory as an Answer -- 1.2.1 Which Theory and Why? -- A Point of History -- Resolvents, Gaussian Tools, and Concentration of Measure Theory -- 1.2.2 The Double Asymptotics: Turning the Curse of Dimensionality into a Dimensionality Blessing -- Why Random Matrix Theory to Study the Large n,p Regime? -- The Case of Machine Learning -- 1.2.3 Analyze, Understand, and Improve Large-Dimensional Machine Learning Methods -- From Low- to Large-Dimensional Intuitions -- Core Random Matrices in Machine Learning Algorithms -- Performance Analysis: Spectral Properties and Functionals |
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Directions of Improvement and New Ideas -- 1.2.4 Exploiting Universality: From Large-Dimensional Gaussian Vectors to Real Data -- Theory versus Practice -- Concentrated Random Vectors and Real Data Modeling -- 1.3 Outline and Online Toolbox -- 1.3.1 Organization of the Book -- 1.3.2 Online Codes -- 2 Random Matrix Theory -- 2.1 Fundamental Objects -- 2.1.1 The Resolvent -- 2.1.2 Spectral Measure and Stieltjes Transform -- 2.1.3 Cauchy's Integral, Linear Eigenvalue Functionals, and Eigenspaces -- 2.1.4 Deterministic and Random Equivalents -- 2.2 Foundational Random Matrix Results |
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2.2.1 Key Lemmas and Identities -- Resolvent Identities -- Perturbation Identities -- Probability Identities -- 2.2.2 The Marčenko-Pastur and Semicircle Laws -- The Marčenko-Pastur Law -- Intuitive idea -- Detailed proof of Theorem 2.4 -- The "Gaussian Method" Alternative -- Wigner Semicircle Law -- 2.2.3 Large-Dimensional Sample Covariance Matrices and Generalized Semicircles -- Large Sample Covariance Matrix Model and its Generalizations -- Generalized Semicircle Law with a Variance Profile -- 2.3 Advanced Spectrum Considerations for Sample Covariances -- 2.3.1 Limiting Spectrum |
| Summary |
This book presents a unified theory of random matrices for applications in machine learning, offering a large-dimensional data vision that exploits concentration and universality phenomena. This enables a precise understanding, and possible improvements, of the core mechanisms at play in real-world machine learning algorithms. The book opens with a thorough introduction to the theoretical basics of random matrices, which serves as a support to a wide scope of applications ranging from SVMs, through semi-supervised learning, unsupervised spectral clustering, and graph methods, to neural networks and deep learning. For each application, the authors discuss small- versus large-dimensional intuitions of the problem, followed by a systematic random matrix analysis of the resulting performance and possible improvements. All concepts, applications, and variations are illustrated numerically on synthetic as well as real-world data, with MATLAB and Python code provided on the accompanying website |
| Bibliography |
Includes bibliographical references and index |
| Notes |
Description based on online resource; title from digital title page (viewed on July 18, 2022) |
| Subject |
Machine learning -- Mathematics
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Matrix analytic methods.
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Matrix analytic methods
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| Form |
Electronic book
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| Author |
Liao, Zhenyu, 1992- author.
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| ISBN |
9781009128490 |
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1009128493 |
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