On information plus noise kernel random matrices

TL;DR

This paper investigates the spectral properties of kernel random matrices derived from high-dimensional data with an 'information plus noise' structure, revealing how noise affects the kernel matrix's behavior and highlighting special properties of the Gaussian kernel.

Contribution

It introduces a new analysis framework for kernel matrices with 'information plus noise' data, including spherical and elliptical noise models, and identifies unique properties of the Gaussian kernel.

Findings

Spectral properties can be approximated by a kernel matrix from the signal data alone.

Gaussian kernel exhibits unique spectral properties in the presence of noise.

Elliptical noise complicates the interpretation of spectral behavior.

Abstract

Kernel random matrices have attracted a lot of interest in recent years, from both practical and theoretical standpoints. Most of the theoretical work so far has focused on the case were the data is sampled from a low-dimensional structure. Very recently, the first results concerning kernel random matrices with high-dimensional input data were obtained, in a setting where the data was sampled from a genuinely high-dimensional structure---similar to standard assumptions in random matrix theory. In this paper, we consider the case where the data is of the type "information${}+{}$noise." In other words, each observation is the sum of two independent elements: one sampled from a "low-dimensional" structure, the signal part of the data, the other being high-dimensional noise, normalized to not overwhelm but still affect the signal. We consider two types of noise, spherical and elliptical. In the spherical setting, we show that the spectral properties of kernel random matrices can be understood from a new kernel matrix, computed only from the signal part of the data, but using (in general) a slightly different kernel. The Gaussian kernel has some special properties in this setting. The elliptical setting, which is important from a robustness standpoint, is less prone to easy interpretation.