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

arXiv:2410.11116v4 Announce Type: replace-cross 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 extbf{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.

Source: arXiv cs.LG | 2026-06-24