Non-asymptotic spectral bounds on the $\varepsilon$-entropy of kernel classes

TL;DR

This paper derives sharp non-asymptotic bounds on the $\

Contribution

It provides new tight upper and lower bounds on the $\\varepsilon$-entropy of kernel classes, improving upon previous results and applicable to various kernel types.

Findings

Bounds are sharp for $p\in [1,2]$ based on eigenvalues.

Bounds depend on Mercer series convergence for $p>2$.

Results are asymptotically tight for specific kernels.

Abstract

Let $K: \boldsymbol{\Omega}\times \boldsymbol{\Omega}$ be a continuous Mercer kernel defined on a compact subset of ${\mathbb R}^n$ and $\mathcal{H}_K$ be the reproducing kernel Hilbert space (RKHS) associated with $K$. Given a finite measure $\nu$ on $\boldsymbol{\Omega}$, we investigate upper and lower bounds on the $\varepsilon$-entropy of the unit ball of $\mathcal{H}_K$ in the space $L_p(\nu)$. This topic is an important direction in the modern statistical theory of kernel-based methods. We prove sharp upper and lower bounds for $p\in [1,+\infty]$. For $p\in [1,2]$, the upper bounds are determined solely by the eigenvalue behaviour of the corresponding integral operator $\phi\to \int_{\boldsymbol{\Omega}} K(\cdot,{\mathbf y})\phi({\mathbf y})d\nu({\mathbf y})$. In constrast, for $p>2$, the bounds additionally depend on the convergence rate of the truncated Mercer series to the kernel $K$ in the $L_p(\nu)$-norm. We discuss a number of consequences of our bounds and show that they are substantially tighter than previous bounds for general kernels. Furthermore, for specific cases, such as zonal kernels and the Gaussian kernel on a box, our bounds are asymptotically tight as $\varepsilon\to +0$.