## 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