The Product of Gaussian Matrices is Close to Gaussian

Yi Li; David P. Woodruff

paper

The Product of Gaussian Matrices is Close to Gaussian

arXiv:2108.09887

Abstract

We study the distribution of the {\it matrix product} $G_{1} G_{2} \dots G_{r}$ of $r$ independent Gaussian matrices of various sizes, where $G_{i}$ is $d_{i - 1} \times d_{i}$ , and we denote $p = d_{0}$ , $q = d_{r}$ , and require $d_{1} = d_{r - 1}$ . Here the entries in each $G_{i}$ are standard normal random variables with mean $0$ and variance $1$ . Such products arise in the study of wireless communication, dynamical systems, and quantum transport, among other places. We show that, provided each $d_{i}$ , $i = 1, \dots, r$ , satisfies $d_{i} \geq C p \cdot q$ , where $C \geq C_{0}$ for a constant $C_{0} > 0$ depending on $r$ , then the matrix product $G_{1} G_{2} \dots G_{r}$ has variation distance at most $δ$ to a $p \times q$ matrix $G$ of i.i.d.\ standard normal random variables with mean $0$ and variance $\prod_{i = 1}^{r - 1} d_{i}$ . Here $δ \to 0$ as $C \to \infty$ . Moreover, we show a converse for constant $r$ that if $d_{i} < C^{'} max {p, q}^{1/2} min {p, q}^{3/2}$ for some $i$ , then this total variation distance is at least $δ^{'}$ , for an absolute constant $δ^{'} > 0$ depending on $C^{'}$ and $r$ . This converse is best possible when $p = Θ (q)$ .

Appears in the Proceedings of APPROX/RANDOM 2021