Spectral Density of Sparse Sample Covariance Matrices

TL;DR

This paper uses statistical mechanics techniques to analyze the eigenvalue density of large sparse sample covariance matrices, highlighting differences from dense matrices especially in the spectrum tail.

Contribution

It introduces a novel application of the replica method to evaluate spectral densities of sparse covariance matrices, emphasizing tail behavior.

Findings

Eigenvalue density differs significantly in the tail region for sparse matrices

Comparison of approximation schemes reveals their effectiveness in tail analysis

Spectral properties of sparse matrices are characterized using statistical mechanics methods

Abstract

Applying the replica method of statistical mechanics, we evaluate the eigenvalue density of the large random matrix (sample covariance matrix) of the form $J = A^{\rm T} A$, where $A$ is an $M \times N$ real sparse random matrix. The difference from a dense random matrix is the most significant in the tail region of the spectrum. We compare the results of several approximation schemes, focusing on the behavior in the tail region.