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

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.