Title: Reconstruction of compact sets using cone fields
Speaker: Katherine Turner (University of Chicago)
Abstract: A standard reconstruction problem is how to discover a compact set from a noisy point cloud that approximates it. When learning manifolds we often use the distance to the closest critical point of the distance function. For more general compact sets we need to use geometric quantities such as $\mu$-critical points that can cope with corners. We prove a sufficient condition, as a bound on the Hausdorff distance between two compact sets, for when certain offsets of these two sets are homotopy equivalent in terms of the absence of $\mu$-critical points in an annular region. We do this by showing the existence of a vector field whose flow provides a deformation retraction. The ambient space can be any Riemannian manifold but we focus on ambient manifolds which have nowhere negative curvature (this includes Euclidean space). In the process, we prove stability theorems for $\mu$-critical points when the ambient space is a manifold.
Seminar URL: https://research.math.osu.edu/tgda/tgda-seminar.html