Convergence of the reach for a sequence of Gaussian

TY - JOUR

T1 - Convergence of the reach for a sequence of Gaussian-embedded manifolds

AU - Adler, Robert J.

AU - Krishnan, Sunder Ram

AU - Taylor, Jonathan E.

AU - Weinberger, Shmuel

N1 - Publisher Copyright: © 2017, Springer-Verlag GmbH Germany.

PY - 2018/8/1

Y1 - 2018/8/1

N2 - 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.

AB - 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.

KW - Asymptotics

KW - Critical radius

KW - Curvature

KW - Fluctuation theory

KW - Gaussian process

KW - Manifold

KW - Random embedding

KW - Reach

UR - http://www.scopus.com/inward/record.url?scp=85029423572&partnerID=8YFLogxK

U2 - 10.1007/s00440-017-0801-1

DO - 10.1007/s00440-017-0801-1

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AN - SCOPUS:85029423572

VL - 171

SP - 1045

EP - 1091

IS - 3-4

ER -