Nonlinear spiked covariance matrices and signal propagation in deep neural networks

Understand

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.