Approximating the Riemannian Metric from Discrete Samples

 Speaker: Barak Sober (Duke University)

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 approximation rates. In this work, we prove that the Riemannian metric of the Moving Least-Squares Manifold (Manifold-MLS) introduced by (Sober & Levin 19) has approximation rates of \(O(h^{k-1})\). In other words, the Manifold-MLS is nearly an isometry with high approximation order. We use this fact to devise an algorithm that computes geodesic distances between points on \(M\) with the same rates of convergence. Finally, we show the potential and the robustness to noise of the proposed method in some numerical simulations.

Time: October 16, 2020 2:30pm-3:30pm
Location: Virtually via Zoom
Host: Wolfgang Dahmen

  Access to the video of the talk