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

arXiv: https://arxiv.org/abs/2011.12478 · HTML: https://mlsanity.com/paper/2011.12478/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Nov 2020.

## Built on by (most cited first, 3 of 3)

- [Inductive Global and Local Manifold Approximation and Projection](https://mlsanity.com/paper/2406.08097/) (2024)
- [Non-Parametric Manifold Learning](https://mlsanity.com/paper/2107.08089/) (2021)
- [Embedding Functional Data: Multidimensional Scaling and Manifold Learning](https://mlsanity.com/paper/2208.14540/) (2022)

## Abstract (arXiv)

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.