Leonid Pastur

Leonid Pastur

The paper deals with the distribution of singular values of the input-output Jacobian of deep untrained neural networks in the limit of their infinite width. The Jacobian is the product of random matrices where the independent rectangular weight matrices alternate with diagonal matrices whose entries depend on the corresponding column of the nearest neighbor weight matrix. The problem was considered in \cite{Pe-Co:18} for the Gaussian weights and biases and also for the weights that are Haar distributed orthogonal matrices and Gaussian biases. Basing on a free probability argument, it was claimed that in these cases the singular value distribution of the Jacobian in the limit of infinite width (matrix size) coincides with that of the analog of the Jacobian with special random but weight independent diagonal matrices, the case well known in random matrix theory. The claim was rigorously proved in \cite{Pa-Sl:21} for a quite general class of weights and biases with i.i.d. (including Gaussian) entries by using a version of the techniques of random matrix theory. In this paper we use another version of the techniques to justify the claim for random Haar distributed weight matrices and Gaussian biases.

Leonid Pastur

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.