Largest Eigenvalues of the Conjugate Kernel of Single-Layered Neural Networks

TL;DR

This paper analyzes the asymptotic behavior of the largest eigenvalues of the conjugate kernel in single-layer neural networks, revealing phase transitions and connections to well-known random matrix models.

Contribution

It establishes the limiting distribution of the largest eigenvalue for nonlinear random matrices from neural networks, linking it to classical models and identifying phase transitions.

Findings

Largest eigenvalue converges to a known limit in probability

Identifies phase transition depending on activation function and data distribution

Connects neural network conjugate kernel eigenvalues to information-plus-noise models

Abstract

This paper is concerned with the asymptotic distribution of the largest eigenvalues for some nonlinear random matrix ensemble stemming from the study of neural networks. More precisely we consider $M= \frac{1}{m} YY^\top$ with $Y=f(WX)$ where $W$ and $X$ are random rectangular matrices with i.i.d. centered entries. This models the data covariance matrix or the Conjugate Kernel of a single layered random Feed-Forward Neural Network. The function $f$ is applied entrywise and can be seen as the activation function of the neural network. We show that the largest eigenvalue has the same limit (in probability) as that of some well-known linear random matrix ensembles. In particular, we relate the asymptotic limit of the largest eigenvalue for the nonlinear model to that of an information-plus-noise random matrix, establishing a possible phase transition depending on the function $f$ and the distribution of $W$ and $X$. This may be of interest for applications to machine learning.