Eigenvalues of block structured asymmetric random matrices

TL;DR

This paper analyzes the eigenvalue distribution of asymmetric block-structured random matrices, providing explicit formulas for spectral radius and density, with applications to neural networks.

Contribution

It introduces a novel analysis of block-structured asymmetric random matrices using Hermitization and Stieltjes transform, deriving explicit spectral properties.

Findings

Spectral radius formula derived

Explicit density function equations provided

Application to neural network models discussed

Abstract

We study the spectrum of an asymmetric random matrix with block structured variances. The rows and columns of the random square matrix are divided into $D$ partitions with arbitrary size (linear in $N$). The parameters of the model are the variances of elements in each block, summarized in $g\in\mathbb{R}^{D\times D}_+$. Using the Hermitization approach and by studying the matrix-valued Stieltjes transform we show that these matrices have a circularly symmetric spectrum, we give an explicit formula for their spectral radius and a set of implicit equations for the full density function. We discuss applications of this model to neural networks.