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 nonrandom 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 socalled macroscopic universality on the considered random matrices.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 DOI:
 10.48550/arXiv.2001.06188
 arXiv:
 arXiv:2001.06188
 Bibcode:
 2020arXiv200106188P
 Keywords:

 Mathematical Physics;
 Statistics  Machine Learning;
 Primary 15B52;
 Secondary 92B20
 EPrint:
 28 pages