Manifold reconstruction and denoising from scattered data in high dimension

In this paper, we present a method for denoising and reconstructing a low-dimensional manifold in a high-dimensional space. We introduce a multidimensional extension of the Locally Optimal Projection algorithm which was proposed by Lipman et al. in 2007 for surface reconstruction in 3D. The high-dimensional generalization bypasses the curse of dimensionality while reconstructing the manifold in high dimension. Given a noisy point-cloud situated near a low dimensional manifold, the proposed solution distributes points near the unknown manifold in a noise-free and quasi-uniformly manner, by leveraging a generalization of the robust L1-median to higher dimensions. We prove that the non-convex computational method converges to a local stationary solution with a bounded linear rate of convergence if the starting point is close enough to the local minimum. The effectiveness of our approach is demonstrated in various numerical experiments, by considering different manifold topologies with various amounts of noise, including a case of a manifold of different co-dimensions at different locations.