Convergence of the reach for a sequence of Gaussian-embedded manifolds. (English) Zbl 1434.60116
Authors’ abstract: 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.
Reviewer: Nikolai N. Leonenko (Cardiff)
MSC:
| 60G15 | Gaussian processes |
| 57N35 | Embeddings and immersions in topological manifolds |
| 60D05 | Geometric probability and stochastic geometry |
| 60G60 | Random fields |
| 60F05 | Central limit and other weak theorems |
| 60G12 | General second-order stochastic processes |
Keywords:
Gaussian process; manifold; random embedding; critical radius; reach; curvature; asymptotics; fluctuation theoryReferences:
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