Extremal Eigenvalues of Random Kernel Matrices with Polynomial Scaling

TL;DR

This paper analyzes the spectral properties of random kernel matrices with polynomial sample scaling, revealing their eigenvalue behavior and convergence to known spectral laws, with implications for high-dimensional data analysis.

Contribution

It provides a decomposition of the kernel matrices into bulk and low-rank parts and establishes eigenvalue convergence results in polynomial scaling regimes.

Findings

Spectral norm of the bulk part converges to the spectrum edge.

Eigenvalues converge to edges of the Marcenko-Pastur law.

Empirical spectral distribution converges to additive free convolution.

Abstract

We study the spectral norm of random kernel matrices with polynomial scaling, where the number of samples scales polynomially with the data dimension. In this regime, Lu and Yau (2022) proved that the empirical spectral distribution converges to the additive free convolution of a semicircle law and a Marcenko-Pastur law. We demonstrate that the random kernel matrix can be decomposed into a "bulk" part and a low-rank part. The spectral norm of the "bulk" part almost surely converges to the edge of the limiting spectrum. In the special case where the random kernel matrices correspond to the inner products of random tensors, the empirical spectral distribution converges to the Marcenko-Pastur law. We prove that the largest and smallest eigenvalues converge to the corresponding spectral edges of the Marcenko-Pastur law.