Nonlinear spiked covariance matrices and signal propagation in deep neural networks

TL;DR

This paper provides a detailed analysis of the eigenvalue and eigenvector structure of nonlinear spiked covariance matrices, especially in neural networks, revealing how low-dimensional signals propagate and are learned through layers.

Contribution

It introduces a precise characterization of signal eigenvalues and eigenvectors in nonlinear spiked covariance models, extending understanding of eigenstructure propagation in neural networks.

Findings

Quantitative description of signal eigenvalue propagation in neural networks.

Analysis of rank-one signal component development during training.

Characterization of eigenvector alignment with target functions.

Abstract

Many recent works have studied the eigenvalue spectrum of the Conjugate Kernel (CK) defined by the nonlinear feature map of a feedforward neural network. However, existing results only establish weak convergence of the empirical eigenvalue distribution, and fall short of providing precise quantitative characterizations of the ''spike'' eigenvalues and eigenvectors that often capture the low-dimensional signal structure of the learning problem. In this work, we characterize these signal eigenvalues and eigenvectors for a nonlinear version of the spiked covariance model, including the CK as a special case. Using this general result, we give a quantitative description of how spiked eigenstructure in the input data propagates through the hidden layers of a neural network with random weights. As a second application, we study a simple regime of representation learning where the weight matrix develops a rank-one signal component over training and characterize the alignment of the target function with the spike eigenvector of the CK on test data.