Eigenvalue distribution of some random matrices

TL;DR

This paper studies the eigenvalue distribution of kernel random matrices with symmetric functions, especially sinc, showing eigenvalues concentrate near 0 and 1, with implications for large parameter regimes.

Contribution

It provides a rigorous analysis of eigenvalue behavior for a class of kernel matrices, including the case of large parameters, extending existing results.

Findings

Eigenvalues are concentrated around 0 and 1.

Eigenvalue frequency near one is proportional to parameter c.

Numerical results support theoretical predictions.

Abstract

In this paper, we investigate the eigenvalue distribution of a class of kernel random matrices whose $(i,j)$-th entry is $f(X_i,X_j)$ where $f$ is a symmetric function belonging to the Paley-Wiener space $\mathcal{B}_c$ and $(X_i)_{1\leq i \leq N}$ are i.i.d. random variables. We rigorously prove that, with high probability, the eigenvalues of these random matrices are well approximated by those of an underlying estimator. A particularly notable case is when $f=sinc$ , which has been widely studied due to its relevance in various scientific fields, including machine learning and telecommunications. In this case, we push forward the general approach by computing the eigenvalues of the estimator. More precisely, we have proved that the eigenvalues are concentrated around zero and one. In particular, we address the case of large values of $c$ with respect to the matrix size $N$, which, to the best of our knowledge, has not been studied in the literature. Furthermore, we establish that the frequency of eigenvalues close to one is proportional to $c$. Numerical results are provided in order to illustrate the theoretical findings.