Which Spaces can be Embedded in $L_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy
This provides a foundational insight into kernel methods for learning complex function spaces, with implications for approximability and learnability in machine learning.
The paper tackles the problem of characterizing which function spaces can be embedded into reproducing kernel Banach spaces by establishing a converse relationship: a bound on metric entropy growth allows embedding into an L_p-type RKBS, showing these spaces model learnable classes with controlled 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.