Universality for the global spectrum of random inner-product kernel matrices in the polynomial regime

TL;DR

This paper demonstrates the universality of the spectral behavior of random inner-product kernel matrices in the polynomial regime, showing that the eigenvalue distribution depends only on the moments of the entries, not their specific distribution.

Contribution

It extends previous results by proving universality of the spectral distribution for a broad class of random matrices in the polynomial regime, including non-integer exponents.

Findings

Eigenvalue distribution is universal for i.i.d. entries with finite moments.

In the polynomial regime, the spectrum is a free convolution of distributions.

For non-integer exponents, the spectrum simplifies to a semicircular law.

Abstract

We consider certain large random matrices, called random inner-product kernel matrices, which are essentially given by a nonlinear function $f$ applied entrywise to a sample-covariance matrix, $f(X^TX)$, where $X \in \mathbb{R}^{d \times N}$ is random and normalized in such a way that $f$ typically has order-one arguments. We work in the polynomial regime, where $N \asymp d^\ell$ for some $\ell > 0$, not just the linear regime where $\ell = 1$. Earlier work by various authors showed that, when the columns of $X$ are either uniform on the sphere or standard Gaussian vectors, and when $\ell$ is an integer (the linear regime $\ell = 1$ is particularly well-studied), the bulk eigenvalues of such matrices behave in a simple way: They are asymptotically given by the free convolution of the semicircular and Mar\v{c}enko-Pastur distributions, with relative weights given by expanding $f$ in the Hermite basis. In this paper, we show that this phenomenon is universal, holding as soon as $X$ has i.i.d. entries with all finite moments. In the case of non-integer $\ell$, the Mar\v{c}enko-Pastur term disappears (its weight in the free convolution vanishes), and the spectrum is just semicircular.