Which Spaces can be Embedded in Lp-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 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.