A Universal Matrix Ensemble that Unifies Eigenspectrum Laws via Neural Network Models

TL;DR

This paper introduces a universal matrix ensemble inspired by neural networks that unifies key eigenspectrum laws in random matrix theory, providing explicit spectral distribution expressions and insights into neural network stability.

Contribution

It establishes a universal matrix ensemble that unifies the Marchenko--Pastur and elliptic laws, deriving explicit spectral distributions and applying them to neural network stability analysis.

Findings

Unified eigenspectrum law for diverse matrix ensembles

Explicit spectral distribution derived from saddle-node analysis

Insights into neural network stability from the universal law

Abstract

Random matrix theory, which characterizes the spectrum distribution of infinitely large matrices, plays a central role in theories across diverse fields, including high-dimensional data analysis, ecology, neuroscience, and machine learning. Among its celebrated achievements, the Marchenko--Pastur law and the elliptic law have served as key results for numerous applications. However, the relationship between these two laws remains elusive, and the existence of a universal framework unifying them is unclear. Inspired by a neural network model, we establish a universal matrix ensemble that unifies these laws as special cases. Through an analysis based on the saddle-node equation, we derive an explicit expression for the spectrum distribution of the ensemble. As a direct application, we reveal how the universal law clarifies the stability of a class of associative memory neural networks. By uncovering a fundamental law of random matrix theory, our results deepen the understanding of high-dimensional systems and advance the integration of theories across multiple disciplines.