Entrywise error bounds for low-rank approximations of kernel matrices

TL;DR

This paper establishes entrywise error bounds for low-rank kernel matrix approximations, providing insights into the statistical behavior of individual entries, supported by theoretical innovations and empirical validation.

Contribution

It introduces novel entrywise error bounds for kernel matrix approximations and a delocalisation result for eigenvectors, advancing understanding beyond spectral norms.

Findings

Derived explicit entrywise error bounds

Established a delocalisation result for eigenvectors

Validated bounds with empirical experiments

Abstract

In this paper, we derive entrywise error bounds for low-rank approximations of kernel matrices obtained using the truncated eigen-decomposition (or singular value decomposition). While this approximation is well-known to be optimal with respect to the spectral and Frobenius norm error, little is known about the statistical behaviour of individual entries. Our error bounds fill this gap. A key technical innovation is a delocalisation result for the eigenvectors of the kernel matrix corresponding to small eigenvalues, which takes inspiration from the field of Random Matrix Theory. Finally, we validate our theory with an empirical study of a collection of synthetic and real-world datasets.