Which Spaces can be Embedded in $L_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy

TL;DR

This paper establishes a new link between metric entropy growth and the embeddability of function spaces into $L_p$-type Reproducing Kernel Banach Spaces, revealing that bounded metric entropy implies such embeddings.

Contribution

It proves a novel converse result showing that bounded metric entropy of a function space guarantees its embedding into $L_p$-type RKBS, broadening the understanding of kernel methods.

Findings

Bounded metric entropy implies embeddability into $L_p$-type RKBS.

Provides a characterization of function spaces suitable for kernel methods.

Enhances understanding of the limitations and capabilities of kernel-based learning.

Abstract

In this paper, we establish a novel connection between the metric entropy growth and the embeddability of function spaces into reproducing kernel Hilbert/Banach spaces. Metric entropy characterizes the information complexity of function spaces and has implications for their approximability and learnability. Classical results show that embedding a function space into a reproducing kernel Hilbert space (RKHS) implies a bound on its metric entropy growth. Surprisingly, we prove a \textbf{converse}: a bound on the metric entropy growth of a function space allows its embedding to a $L_p-$type Reproducing Kernel Banach Space (RKBS). This shows that the ${L}_p-$type RKBS provides a broad modeling framework for learnable function classes with controlled metric entropies. Our results shed new light on the power and limitations of kernel methods for learning complex function spaces.