# Approximating the Riemannian Metric from Point Clouds via Manifold Moving Least Squares

arXiv: https://arxiv.org/abs/2007.09885 · HTML: https://mlsanity.com/paper/2007.09885/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 20 Jul 2020.

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

- [Non-Parametric Estimation of Manifolds from Noisy Data](https://mlsanity.com/paper/2105.04754/) (2021)
- [Manifold Free Riemannian Optimization](https://mlsanity.com/paper/2209.03269/) (2022)
- [Fast, Accurate Manifold Denoising by Tunneling Riemannian Optimization](https://mlsanity.com/paper/2502.16819/) (2025)

## Abstract (arXiv)

The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold M of Euclidean space has been a long-standing challenge in computational geometry. Given a sampling resolution parameter  h , state-of-the-art discrete methods yield  O(h)  provable approximations. In this paper, we investigate the convergence of such approximations made by Manifold Moving Least-Squares (Manifold-MLS), a method that constructs an approximating manifold Mh using information from a given point cloud that was developed by Sober & Levin in 2019. In this paper, we show that provided that M∈ Ck and closed (i.e. M is a compact manifold without boundary) the Riemannian metric of  Mh  approximates the Riemannian metric of  M, . Explicitly, given points  p1, p2 ∈ M with geodesic distance  ρM(p1, p2) , we show that their corresponding points  p1h, p2h ∈ Mh have a geodesic distance of  ρMh(p1h,p2h) = ρM(p1, p2)(1 + O(hk-1))  (i.e., the Manifold-MLS is nearly an isometry). We then use this result, as well as the fact that  Mh  can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with a rate of convergence  O(hk-1) . We show the potential and the robustness to noise of the proposed method on some numerical simulations.