On Random Matrices Arising in Deep Neural Networks. Gaussian Case

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.

28 pages

References in corpus (1)

• Deep Learning in Neural Networks: An Overview

Cited by in corpus (9)

• Tensor Programs III: Neural Matrix Laws
• A Random Matrix Analysis of Random Fourier Features: Beyond the Gaussian Kernel, a Precise Phase Transition, and the Corresponding Double Descent
• Overparameterized Linear Regression under Adversarial Attacks
• Conditioning of Random Feature Matrices: Double Descent and Generalization Error
• Universal characteristics of deep neural network loss surfaces from random matrix theory
• Eigenvalue Distribution of Large Random Matrices Arising in Deep Neural Networks: Orthogonal Case
• The Spectrum of Fisher Information of Deep Networks Achieving Dynamical Isometry
• Linear Eigenvalue Statistics of $X{X}^{\prime }$ matrices
• Large-Dimensional Random Matrix Theory and Its Applications in Deep Learning and Wireless Communications