Nonlinear spiked covariance matrices and signal propagation in deep neural networks
This work addresses a gap in theoretical understanding for researchers in deep learning theory, offering incremental insights into signal propagation and representation learning.
The authors tackled the problem of characterizing spike eigenvalues and eigenvectors in nonlinear spiked covariance models, including neural network Conjugate Kernels, and provided quantitative descriptions of how signal structure propagates through deep networks and aligns during representation learning.
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.