On Random Matrices Arising in Deep Neural Networks. Gaussian Case

TL;DR

This paper analyzes the distribution of singular values of products of Gaussian random matrices in deep neural networks, providing theoretical justification for free probability approaches and extending results to broader data distributions.

Contribution

It offers a rigorous justification for using free probability in analyzing neural network matrices with Gaussian data and extends universality results to non-Gaussian data.

Findings

Distribution of singular values characterized for Gaussian data matrices.

Justification of free probability methods in this context.

Extension of universality to non-Gaussian data with finite moments.

Abstract

The paper deals with distribution of singular values of product of random matrices arising in the analysis of deep neural networks. The matrices resemble the product analogs of the sample covariance matrices, however, an important difference is that the population covariance matrices, which are assumed to be non-random in the standard setting of statistics and random matrix theory, are now random, moreover, are certain functions of random data matrices. The problem has been considered in recent work [21] by using the techniques of free probability theory. Since, however, free probability theory deals with population matrices which are independent of the data matrices, its applicability in this case requires an additional justification. We present this justification by using a version of the standard techniques of random matrix theory under the assumption that the entries of data matrices are independent Gaussian random variables. In the subsequent paper [18] we extend our results to the case where the entries of data matrices are just independent identically distributed random variables with several finite moments. This, in particular, extends the property of the so-called macroscopic universality on the considered random matrices.