The approximation of both geodesic distances and shortest paths on point cloud sampled from an embedded submanifold $\mathcal{ M}$ 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 $\mathcal{ M}^h$ using information from a given point cloud that was developed by Sober \& Levin in 2019. In this paper, we show that provided that $\mathcal{ M}\in C^{ k}$ and closed (i.e. $\mathcal{ M}$ is a compact manifold without boundary) the Riemannian metric of $ \mathcal{ M}^h $ approximates the Riemannian metric of $ \mathcal{ M}, $. Explicitly, given points $ p_1, p_2 \in \mathcal{ M}$ with geodesic distance $ \rho_{ \mathcal{ M}}(p_1, p_2) $, we show that their corresponding points $ p_1^h, p_2^h \in \mathcal{ M}^h$ have a geodesic distance of $ \rho_{ \mathcal{ M}^h}(p_1^h,p_2^h) = \rho_{ \mathcal{ M}}(p_1, p_2)(1 + O(h^{ k-1})) $ (i.e., the Manifold-MLS is nearly an isometry). We then use this result, as well as the fact that $ \mathcal{ M}^h $ can be sampled with any desired resolution, to devise a naive algorithm that yields approximate geodesic distances with a rate of convergence $ O(h^{ k-1}) $. We show the potential and the robustness to noise of the proposed method on some numerical simulations.
Categories: math.NA cs.CG cs.LG cs.NA