Which Spaces can be Embedded in $L_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy (2410.11116v2)

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 \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.

• The paper establishes that a bound on metric entropy growth enables embedding of learnable function classes into ℒp-type RKBSs.
• The authors employ advanced techniques like Dudley’s chaining and type-cotype theory to rigorously derive embeddability conditions.
• The findings offer a practical framework for kernel methods in machine learning by simplifying metric entropy estimation.

Embedding Spaces into $\mathcal{L}_p$-Type Reproducing Kernel Banach Spaces

The paper "Which Spaces can be Embedded in $\mathcal{L}_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy" offers a thorough examination of the relationship between metric entropy and the embeddability of function spaces into Reproducing Kernel Banach Spaces (RKBSs). The authors aim to address whether $\mathcal{L}_p$-type RKBSs can serve as an adequate framework for machine learning, focusing on the embeddability of spaces based on their metric entropy properties.

Key Contributions and Results

1. Connection between Metric Entropy and Embeddability: The paper establishes that a bound on the metric entropy growth of a function space implies its embedding into a $\mathcal{L}_p$-type RKBS. This is a significant extension of classical results which only showed that embedding into a Reproducing Kernel Hilbert Space (RKHS) puts a bound on metric entropy growth.
2. Theoretical Framework: The paper provides a characterization that links the embeddability of function spaces to their learnability by utilizing metric entropy. Specifically, the paper demonstrates that every function class that is learnable with a polynomial number of data points can be embedded into a $\mathcal{L}_p$-type RKBS, contingent on controlling the metric entropy growth.
3. Implications for Machine Learning: The results underscore that $\mathcal{L}_p$-type RKBSs offer an expressive model class for learnable function classes. The findings are particularly beneficial as they simplify the estimation of metric entropy, which is often easier than directly assessing embeddability.
4. Applications and Examples: The paper provides applications of its main theorem to well-known function spaces, including spaces of mixed smoothness and Barron spaces. Corollaries derived from the main result demonstrate the embeddability of specific Besov spaces into $\mathcal{L}_p$-type RKBSs, demonstrating the breadth of applicability of their theorem.

Methodological Insights

The authors employ a blend of sophisticated mathematical tools to achieve their results. Notably:

• Dudley's Chaining for Banach Spaces: A generalization of classic chaining techniques to abstract Banach spaces is utilized to establish bounds on the Rademacher norm.
• Type and Cotype Theory: The exploration of the type and cotype of Banach spaces is pivotal in establishing conditions under which spaces can be embedded in $\mathcal{L}_p$-type RKBSs.
• Hahn-Banach Theorem Application: By leveraging Hahn-Banach extension, the authors creatively construct feature maps necessary for embedding certain function spaces into RKBSs.

Implications and Future Directions

The paper's implications are twofold: practically, it provides a framework for embedding machine learning-relevant spaces into $\mathcal{L}_p$-type RKBSs, thus broadening the scope of applicable kernel methods; theoretically, it advances the understanding of the relationship between metric entropy and functional space embeddability.

Future research may explore extending these results to broader classes of function spaces or examining the impact of specific conditions on metric entropy, which could provide further insight into the limitations and capabilities of $\mathcal{L}_p$-type RKBSs in modeling complex, high-dimensional data spaces. Additionally, investigating the computational aspects of these embeddings could yield practical algorithms for machine learning applications.

In conclusion, this paper presents a rigorous and detailed paper of a fundamental problem in functional analysis and machine learning, providing valuable insights into the embeddability of function spaces into $\mathcal{L}_p$-type RKBSs through the lens of metric entropy.