It is common to assume that the data was sampled from a low-dimensional manifold in a high-dimensional space. In real life, neither the dimension of this manifold nor its geometry is known, and the data is often contaminated with noise and outliers. In this talk, we first present a 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-uniform 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. Next, we assume that the scatter data has its own geometry, and model the data collection as a (nonlinear) fibre bundle with a connection. We then focus on examining the geometric properties of both the base manifold and the fiber bundles. We demonstrate our methodology on manifolds of various dimensions, as well as on a collection of anatomical surfaces, and aim to shed light on questions in evolutionary anthropology.

Based on a joined works with David Levin, Shan Shan, Rui Xin, Alex Winn, and Ingrid Daubechies