  In this work a novel method to quantify spectral ergodicity for random
matrices is presented. The new methodology combines approaches rooted in the
metrics of Thirumalai-Mountain (TM) and Kullbach-Leibler (KL) divergence. The
method is applied to a general study of deep and recurrent neural networks via
the analysis of random matrix ensembles mimicking typical weight matrices of
those systems. In particular, we examine circular random matrix ensembles:
circular unitary ensemble (CUE), circular orthogonal ensemble (COE), and
circular symplectic ensemble (CSE). Eigenvalue spectra and spectral ergodicity
are computed for those ensembles as a function of network size. It is observed
that as the matrix size increases the level of spectral ergodicity of the
ensemble rises, i.e., the eigenvalue spectra obtained for a single realisation
at random from the ensemble is closer to the spectra obtained averaging over
the whole ensemble. Based on previous results we conjecture that success of
deep learning architectures is strongly bound to the concept of spectral
ergodicity. The method to compute spectral ergodicity proposed in this work
could be used to optimise the size and architecture of deep as well as
recurrent neural networks.
