From Geometry to Topology: Inverse Theorems for Distributed Persistence
Speaker: Elchanan Solomon (Duke University)
Occasion: Topological Data Analysis
Date: April 26, 2021
Abstract: What is the “right” topological invariant of a large point cloud X? Prior research has focused on estimating the full persistence diagram of X, a quantity that is very expensive to compute, unstable to outliers, and far from a sufficient statistic. We therefore propose that the correct invariant is not the persistence diagram of X, but rather the collection of persistence diagrams of many small subsets. This invariant, which we call “distributed persistence,” is trivially parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of point clouds (with the quasi-isometry metric) to the space of distributed persistence invariants (with the Hausdorff-Bottleneck distance) is a global quasi-isometry. This is a much stronger property than simply being a sufficient statistic, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the quasi-isometry bounds depend on the size of the subsets taken, so that as the size of these subsets goes from small to large, the invariant interpolates between a purely geometric one and a purely topological one. Lastly, we note that our inverse results do not actually require considering all subsets of a fixed size (an enormous collection), but a relatively small collection satisfying certain covering properties, properties that arise with high probability when randomly selecting sufficiently many subsets. These theoretical results are complemented by experiments showcasing the success of distributed persistence at solving a number of morphometric challenges. This is joint work with Alex Wagner and Paul Bendich.