Statistics > Machine Learning
[Submitted on 9 Apr 2022 (v1), last revised 30 Dec 2024 (this version, v2)]
Title:Non-asymptotic spectral bounds on the $\varepsilon$-entropy of kernel classes
View PDF HTML (experimental)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 $\varepsilon\to +0$.
Submission history
From: Rustem Takhanov [view email][v1] Sat, 9 Apr 2022 16:45:22 UTC (48 KB)
[v2] Mon, 30 Dec 2024 17:41:16 UTC (60 KB)
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