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

TL;DR

This paper introduces a minimax optimal method for estimating intrinsic distances on smooth manifolds by surface reconstruction, specifically using the tangential Delaunay complex, and applies this to improve Isomap in the isometric setting.

Contribution

It proposes a minimax optimal approach for intrinsic distance estimation via surface reconstruction and extends this to enhance Isomap for isometric manifold learning.

Findings

Minimax optimality achieved through surface reconstruction.

Tangential Delaunay complex effectively reconstructs the surface.

Enhanced Isomap method for isometric manifold learning.

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.