Random matrix methods for machine learning

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.

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

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

Directions of Improvement and…