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

2024-10-14

AI Summary

This work addresses the necessary and sufficient conditions for embedding a function space into an $L_p$-type reproducing kernel Banach space (RKBS). Method: By integrating functional analysis, metric entropy theory, and the geometric structure of RKBSs, the authors establish an exact characterization linking embeddability to the growth rate of metric entropy. Contribution/Results: They prove that a function space embeds into an $L_p$-type RKBS if and only if the metric entropy of its unit ball satisfies a specific upper bound—crucially, this bound alone is sufficient to guarantee existence of such an embedding. This reveals the universal modeling capacity of $L_p$-type RKBSs for function classes exhibiting controlled entropy growth. The result unifies and extends the theoretical scope of classical kernel methods, providing a novel analytical framework for learning high-dimensional, nonsmooth, and low-regularity functions.

Technology Category

• Machine Learning: Kernel Methods
• Knowledge Representation and Reasoning: Computational Complexity of Reasoning
• Computer Vision: Representation Learning for Vision

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Semantics and Knowledge: Methods to enhance, augment, integrate or synergize semantic models such as knowledge graphs and LLMs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

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.

Problem

• Characterizing embeddable spaces via metric entropy growth
• Establishing converse embedding results for L_p-type RKBS
• Determining learnable function classes with controlled entropy

Innovation

• Metric entropy bound enables embedding into RKBS
• L_p-type RKBS models learnable function classes
• Novel connection between entropy growth and embeddability