Eigenvalue Distribution of Large Random Matrices Arising in Deep Neural Networks: Orthogonal Case

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.