Spectral distribution of sparse Gaussian Ensembles of Real Asymmetric Matrices

TL;DR

This paper develops a unified mathematical framework to analyze the spectral properties of sparse real asymmetric Gaussian matrix ensembles, revealing universal behaviors across different sparsity structures relevant to neural networks.

Contribution

It introduces a complexity parameter approach to derive spectral densities for various sparse real asymmetric ensembles, generalizing previous models and uncovering universality.

Findings

Derived spectral densities for real and complex eigenvalues.

Established a universal mathematical formulation for diverse sparse ensembles.

Demonstrated the applicability to neural network modeling.

Abstract

Theoretical analysis of biological and artificial neural networks e.g. modelling of synaptic or weight matrices necessitate consideration of the generic real-asymmetric matrix ensembles, those with varying order of matrix elements e.g. a sparse structure or a banded structure. We pursue the complexity parameter approach to analyze the spectral statistics of the multiparametric Gaussian ensembles of real asymmetric matrices and derive the ensemble averaged spectral densities for real as well as complex eigenvalues. Considerations of the matrix elements with arbitrary choice of mean and variances render us the freedom to model the desired sparsity in the ensemble. Our formulation provides a common mathematical formulation of the spectral statistics for a wide range of sparse real-asymmetric ensembles and also reveals, thereby, a deep rooted universality among them.