Dr. Shira Faigenbaum, Duke University – The MLOP framework: reconstructing a manifold from noisy data in high-dimension

Mode: In-Person

Title: The MLOP framework: reconstructing a manifold from noisy data in high-dimension

Abstract: In this talk, we present the Manifold Locally Optimal Projection (MLOP) method for denoising and reconstructing a low-dimensional manifold in a high-dimensional space. 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 method is demonstrated by reconstructing different manifold topologies with various amounts of noise. Subsequently, we discuss two extensions of the MLOP framework to deal with hole-filling in manifolds, and the approximation of functions over manifolds.