Authors
Robert J Adler, Sunder Ram Krishnan, Jonathan E Taylor, Shmuel Weinberger
Publication date
2018/8
Journal
Probability Theory and Related Fields
Volume
171
Pages
1045-1091
Publisher
Springer Berlin Heidelberg
Description
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.
Scholar articles
RJ Adler, SR Krishnan, JE Taylor, S Weinberger - Probability Theory and Related Fields, 2018