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.

本研究针对出现在深度神经网络分析中的随机矩阵乘积奇异值分布进行了研究，其中，数据矩阵的总体协方差矩阵是随机的，基于随机矩阵理论和标准技术，分析了数据矩阵的非高斯分布并阐述其在分析宏观普适性方面的潜在应用。