Reproducing Kernel Hilbert Spaces and entropy Kolmogorov numbers on compact Lie Groups

TL;DR

This work studies the entropy (covering) numbers for the embedding of the reproducing kernel Hilbert space $\mathcal{H}_K$ into $C(G)$ on a compact Lie group $G$, where $K$ is the kernel of a left-invariant, trace-class operator $T$ on $L^2(G)$. By exploiting group Fourier analysis and the symbol calculus for invariant operators, the authors provide an explicit RKHS description and unitary correspondence with a sequence space, enabling precise operator-norm and rank estimates for truncated Fourier projections. They derive sharp upper and lower bounds on the covering numbers $\mathcal{C}(\varepsilon,I_K)$ in terms of the trace and determinant orders $\beta$ and $\gamma$ of the symbol $\sigma_T$, yielding rates like $\varepsilon^{-2n/(\beta-n)}$ (upper) and $(\log(1/\varepsilon))^{1+n/\gamma}$ (lower). These results extend entropy/Kolmogorov-number analyses from intervals and homogeneous spaces to the setting of compact Lie groups, with potential implications for kernel-based approximation and learning on manifolds. The approach integrates harmonic analysis on groups, Schatten-class operator theory, and pseudo-differential symbol methods to quantify the complexity of RKHS embeddings on $G$.

Abstract

On a compact Lie group $G$, we consider the reproducing kernel Hilbert space $\mathcal{H}_K$ associated with the integral kernel $K$ of a left-invariant, positive, symmetric, trace class integral operator on $L^2(G)$. We present lower and upper asymptotic estimates for the entropy Kolmogorov numbers (also called covering numbers) for the embedding of $\mathcal{H}_K$ into the space $C(G)$ of continuous functions on $G$.

Theorems & Definitions (13)

• Proposition 2.1
• Lemma 3.1
• Lemma 3.2
• Theorem 3.3
• Theorem 3.4
• Remark 4.1
• Theorem 4.2
• Lemma 5.1
• Remark 5.2
• Definition 5.3