A First Course in Random Matrix Theory: for Physicists, Engineers and Data Scientists

✍ Scribed by Marc Potters

• Publisher: Cambridge University Press
• Year: 2020
• Tongue: English
• Leaves: 371
• Category: Library

✦ Synopsis

The real world is perceived and broken down as data, models and algorithms in the eyes of physicists and engineers. Data is noisy by nature and classical statistical tools have so far been successful in dealing with relatively smaller levels of randomness. The recent emergence of Big Data and the required computing power to analyse them have rendered classical tools outdated and insufficient. Tools such as random matrix theory and the study of large sample covariance matrices can efficiently process these big data sets and help make sense of modern, deep learning algorithms. Presenting an introductory calculus course for random matrices, the book focusses on modern concepts in matrix theory, generalising the standard concept of probabilistic independence to non-commuting random variables. Concretely worked out examples and applications to financial engineering and portfolio construction make this unique book an essential tool for physicists, engineers, data analysts, and economists.

✦ Table of Contents

Copyright
Contents
Preface
List of Symbols
Part I Classical Random Matrix Theory
1 Deterministic Matrices
1.1 Matrices, Eigenvalues and Singular Values
1.2 Some Useful Theorems and Identities
2 Wigner Ensemble and Semi-Circle Law
2.1 Normalized Trace and Sample Averages
2.2 The Wigner Ensemble
2.3 Resolvent and Stieltjes Transform
3 More on Gaussian Matrices
3.1 Other Gaussian Ensembles
3.2 Moments and Non-Crossing Pair Partitions
4 Wishart Ensemble and Marˇ cenko–Pastur Distribution
4.1 Wishart Matrices
4.2 Marˇ cenko–Pastur Using the Cavity Method
5 Joint Distribution of Eigenvalues
5.1 From Matrix Elements to Eigenvalues
5.2 Coulomb Gas and Maximum Likelihood Configurations
5.3 Applications: Wigner, Wishart and the One-Cut Assumption
5.4 Fluctuations Around the Most Likely Configuration
5.5 An Eigenvalue Density Saddle Point
6 Eigenvalues and Orthogonal Polynomials
6.1 Wigner Matrices and Hermite Polynomials
6.2 Laguerre Polynomials
6.3 Unitary Ensembles
7 The Jacobi Ensemble
7.1 Properties of Jacobi Matrices
7.2 Jacobi Matrices and Jacobi Polynomials
Part II Sums and Products of Random Matrices
8 Addition of Random Variables and Brownian Motion
8.1 Sums of Random Variables
8.2 Stochastic Calculus
9 Dyson Brownian Motion
9.1 Dyson Brownian Motion I: Perturbation Theory
9.2 Dyson Brownian Motion II: Itoˆ Calculus
9.3 The Dyson Brownian Motion for the Resolvent
9.4 The Dyson Brownian Motion with a Potential
9.5 Non-Intersecting Brownian Motions and the Karlin–McGregor Formula
10 Addition of Large Random Matrices
10.1 Adding a Large Wigner Matrix to an Arbitrary Matrix
10.2 Generalization to Non-Wigner Matrices
10.3 The Rank-1 HCIZ Integral
10.4 Invertibility of the Stieltjes Transform
10.5 The Full-Rank HCIZ Integral
11 Free Probabilities
11.1 Algebraic Probabilities: Some Definitions
11.2 Addition of Commuting Variables
11.3 Non-Commuting Variables
11.4 Free Product
12 Free Random Matrices
12.1 Random Rotations and Freeness
12.2 R-Transforms and Resummed Perturbation Theory
12.3 The Central Limit Theorem for Matrices
12.4 Finite Free Convolutions
12.5 Freeness for 2 × 2 Matrices
13 The Replica Method
13.1 Stieltjes Transform
13.2 Resolvent Matrix
13.3 Rank-1 HCIZ and Replicas
13.4 Spin-Glasses, Replicas and Low-Rank HCIZ
14 Edge Eigenvalues and Outliers
14.1 The Tracy–Widom Regime
14.2 Additive Low-Rank Perturbations
14.3 Fat Tails
14.4 Multiplicative Perturbation
14.5 Phase Retrieval and Outliers
Part III Applications
15 Addition and Multiplication: Recipes and Examples
15.1 Summary
15.2 R- and S-Transforms and Moments of Useful Ensembles
15.3 Worked-Out Examples: Addition
15.4 Worked-Out Examples: Multiplication
16 Products of Many Random Matrices
16.1 Products of Many Free Matrices
16.2 The Free Log-Normal
16.3 A Multiplicative Dyson Brownian Motion
16.4 The Matrix Kesten Problem
17 Sample Covariance Matrices
17.1 Spatial Correlations
17.2 Temporal Correlations
17.3 Time Dependent Variance
17.4 Empirical Cross-Covariance Matrices
18 Bayesian Estimation
18.1 Bayesian Estimation
18.2 Estimating a Vector: Ridge and LASSO
18.3 Bayesian Estimation of the True Covariance Matrix
19 Eigenvector Overlaps and Rotationally Invariant Estimators
19.1 Eigenvector Overlaps
19.2 Rotationally Invariant Estimators
19.3 Properties of the Optimal RIE for Covariance Matrices
19.4 Conditional Average in Free Probability
19.5 Real Data
19.6 Validation and RIE
20 Applications to Finance
20.1 Portfolio Theory
20.2 The High-Dimensional Limit
20.3 The Statistics of Price Changes: A Short Overview
20.4 Empirical Covariance Matrices
Appendix Mathematical Tools
A.1 Saddle Point Method
A.2 Tricomi’s Formula
A.3 Toeplitz and Circulant Matrices
Index