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

2020/11/25 by Arias-Castro, Ery, Chau, Phong Alain
#FOS: Computer and information sciences #FOS: Mathematics #Machine Learning (cs.LG) #Machine Learning (stat.ML) #Statistics Theory (math.ST)

paper · doi:10.48550/arxiv.2011.12478

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.

Related

• Minimax adaptive estimation in manifold inference
• Sample-Efficient Geometry Reconstruction from Euclidean Distances using Non-Convex Optimization
• Stability and Minimax Optimality of Tangential Delaunay Complexes for Manifold Reconstruction
• Rehabilitating Isomap: Euclidean Representation of Geodesic Structure
• Minimax Manifold Estimation