The spectrum of kernel random matrices

TL;DR

This paper analyzes the spectral properties of kernel random matrices in high-dimensional settings, revealing that their behavior simplifies to linear models, challenging common heuristics in statistical and machine learning applications.

Contribution

It provides a rigorous analysis of kernel matrix spectra in high dimensions, showing a surprising linearization effect and questioning their practical modeling relevance.

Findings

Kernel matrices behave linearly in high dimensions.

Surprising simplification contrasts with heuristic expectations.

Raises questions about the applicability of random matrix models.

Abstract

We place ourselves in the setting of high-dimensional statistical inference where the number of variables $p$ in a dataset of interest is of the same order of magnitude as the number of observations $n$. We consider the spectrum of certain kernel random matrices, in particular $n\times n$ matrices whose $(i,j)$th entry is $f(X_i'X_j/p)$ or $f(\Vert X_i-X_j\Vert^2/p)$ where $p$ is the dimension of the data, and $X_i$ are independent data vectors. Here $f$ is assumed to be a locally smooth function. The study is motivated by questions arising in statistics and computer science where these matrices are used to perform, among other things, nonlinear versions of principal component analysis. Surprisingly, we show that in high-dimensions, and for the models we analyze, the problem becomes essentially linear--which is at odds with heuristics sometimes used to justify the usage of these methods. The analysis also highlights certain peculiarities of models widely studied in random matrix theory and raises some questions about their relevance as tools to model high-dimensional data encountered in practice.