Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting

Ery Arias-Castro; Phong Alain Chau

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arXiv · 2011.12478

Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting

Ery Arias-Castro·Phong Alain Chau

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Abstract

We start by considering the problem of estimating intrinsic distances on a smooth submanifold. We show that minimax optimality can be obtained via a reconstruction of the surface, and discuss the use of a particular mesh construction -- the tangential Delaunay complex -- for that purpose. We then turn to manifold learning and argue that a variant of Isomap where the distances are instead computed on a reconstructed surface is minimax optimal for the isometric variant of the problem.

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BibTeX RIS

Ery Arias-Castro, Phong Alain Chau. 2020-11-25. Minimax Estimation of Distances on a Surface and Minimax Manifold Learning in the Isometric-to-Convex Setting. https://arxiv.org/abs/2011.12478

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