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

arXiv: https://arxiv.org/abs/2410.11116 · HTML: https://mlsanity.com/paper/2410.11116/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- First posted on arXiv 14 Oct 2024.

## Abstract (arXiv)

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 converse: a bound on the metric entropy growth of a function space allows its embedding to a Lp-type Reproducing Kernel Banach Space (RKBS). This shows that the Lp-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.