Non-asymptotic Results for Singular Values of Gaussian Matrix Products

TL;DR

This paper provides non-asymptotic probabilistic bounds and convergence rates for the singular values and Lyapunov exponents of products of Gaussian matrices, extending classical ergodic theory results to finite regimes.

Contribution

It introduces non-asymptotic concentration estimates and convergence rates for singular values and Lyapunov exponents of Gaussian matrix products, using novel small ball probability techniques.

Findings

Concentration estimates for sums of Lyapunov exponents

Quantitative convergence rate of empirical singular value distribution

Joint normality of Lyapunov exponents for large N

Abstract

This article concerns the non-asymptotic analysis of the singular values (and Lyapunov exponents) of Gaussian matrix products in the regime where $N,$ the number of term in the product, is large and $n,$ the size of the matrices, may be large or small and may depend on $N$. We obtain concentration estimates for sums of Lyapunov exponents, a quantitative rate of convergence of the empirical measure of the normalized squared singular values to the uniform distribution on $[0,1]$, and results on the joint normality of Lyapunov exponents when $N$ is sufficiently large as a function of $n.$ Our technique consists of non-asymptotic versions of the ergodic theory approach at $N=\infty$ due originally to Furstenberg and Kesten in the 1960's, which were then further developed by Newman and Isopi-Newman as well as by a number of other authors in the 1980's. Our key technical idea is that small ball probabilities for volumes of random projections give a way to quantify convergence in the multiplicative ergodic theorem for random matrices.