Towards Empirical Process Theory for Vector-Valued Functions: Metric Entropy of Smooth Function Classes

2022/02/09 by Jun-Hyung Park, Park, Junhyung, Krikamol Muandet +1 · 3 citations
Computer Science · Physics and Astronomy · #FOS: Mathematics #Neural Networks and Applications #Statistical Mechanics and Entropy #Statistical and Computational Modeling #Statistics Theory (math.ST)

paper · pdf · doi:10.48550/arxiv.2202.04415

openalex publication_date 2022/02/09 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

This paper provides some first steps in developing empirical process theory for functions taking values in a vector space. Our main results provide bounds on the entropy of classes of smooth functions taking values in a Hilbert space, by leveraging theory from differential calculus of vector-valued functions and fractal dimension theory of metric spaces. We demonstrate how these entropy bounds can be used to show the uniform law of large numbers and asymptotic equicontinuity of the function classes, and also apply it to statistical learning theory in which the output space is a Hilbert space. We conclude with a discussion on the extension of Rademacher complexities to vector-valued function classes.

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