Convergence of the reach for a sequence of Gaussian-embedded manifolds

• 成果类型: Article
• 署名作者: Adler, Robert J.; Krishnan, Sunder Ram; Taylor, Jonathan E.; Weinberger, Shmuel
• 署名单位: Technion Israel Institute of Technology; Stanford University; University of Chicago
• 刊物名称: PROBABILITY THEORY AND RELATED FIELDS
• ISSN/ISSBN: 0178-8051
• DOI: 10.1007/s00440-017-0801-1
• 发表日期: 2018
• 页码: 1045-1091
• 关键词: tubes; reconstruction; volume; noisy

摘要

Motivated by questions of manifold learning, we study a sequence of random manifolds, generated by embedding a fixed, compact manifold M into Euclidean spheres of increasing dimension via a sequence of Gaussian mappings. One of the fundamental smoothness parameters of manifold learning theorems is the reach, or critical radius, of M. Roughly speaking, the reach is a measure of a manifold's departure from convexity, which incorporates both local curvature and global topology. This paper develops limit theory for the reach of a family of random, Gaussian-embedded, manifolds, establishing both almost sure convergence for the global reach, and a fluctuation theory for both it and its local version. The global reach converges to a constant well known both in the reproducing kernel Hilbert space theory of Gaussian processes, as well as in their extremal theory.