When examining datasets acquired in data-driven applications, a common assumption is 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 will start by presenting 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. Next, we will examine the scenario when the scatter data has its own geometry, and we wish to study the manifolds of manifolds. Adopting the terminology from differential geometry we examine 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.
