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

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## 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.

## Keywords

Gaussian process Manifold Random embedding Critical radius Reach Curvature Asymptotics Fluctuation theory## Mathematics Subject Classification

Primary 60G15 57N35 Secondary 60D05 60G60## Notes

### Acknowledgements

We would like to thank Takashi Owada for useful discussions, and two referees for helpful comments.

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