揭示函数空间可嵌入L_p型核空间的度量熵条件，补全了核方法理论的关键拼图。
Which Spaces can be Embedded in $L_p$-type Reproducing Kernel Banach Space? A Characterization via Metric Entropy
- 通过度量熵增长约束，证明函数空间可嵌入L_p型再生核巴拿赫空间
- 首次建立度量熵有界与可嵌入性之间的双向联系，突破传统单向结论
- 为复杂函数类学习提供统一框架，适合研究核方法理论与泛化性能的学者
本文建立了度量熵增长与函数空间嵌入再生核希尔伯特/巴拿赫空间之间的新关联。度量熵刻画函数空间的信息复杂度，影响其可逼近性与可学习性。经典结果表明，函数空间嵌入再生核希尔伯特空间（RKHS）会带来度量熵增长的上界。令人惊讶的是，我们证明了其反向成立：若函数空间的度量熵增长有界，则可嵌入至$L_p$-型再生核巴拿赫空间（RKBS）。这一发现表明，$L_p$-型RKBS为具有可控度量熵的学习函数类提供了广泛建模框架。研究成果深化了对核方法在复杂函数空间学习中能力与局限性的理解。
原文摘要 · Abstract (English)
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.
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