Eigen Analysis of Conjugate Kernel and Neural Tangent Kernel

In this paper, we investigate deep feedforward neural networks with random weights. The input data matrix $\boldsymbol{X}$ is drawn from a Gaussian mixture model. We demonstrate that certain eigenvalues of the conjugate kernel and neural tangent kernel may lie outside the support of their limiting spectral measures in the high-dimensional regime. The existence and asymptotic positions of such isolated eigenvalues are rigorously analyzed. Furthermore, we provide a precise characterization of the entrywise limit of the projection matrix onto the eigenspace associated with these isolated eigenvalues. Our findings reveal that the eigenspace captures inherent group features present in $\boldsymbol{X}$. This study offers a quantitative analysis of how group features from the input data evolve through hidden layers in randomly weighted neural networks.