A Higher-Dimensional Homologically Persistent Skeleton

27^th February 2018, 13:00 Ashton Lecture Theater
Dr. Sara Kalisnik
Max Planck Institute for Mathematics in the Sciences

Abstract

A data set is often given as a point cloud, i.e. a non-empty finite metric space. An important problem is to detect the topological shape of data
– for example, to approximate a point cloud by a low-dimensional non-linear subspace such as a graph or a simplicial complex. Classical clustering methods and principal component analysis work very well when data points split into well-separated groups or lie near linear subspaces. Methods from topological data analysis detect more complicated patterns such as holes and voids that persist for a long time in a 1-parameter family of shapes associated to a point cloud. V. Kurlin suggested representing them in a form of a 1-dimensional homologically persistent skeleton, which optimally extends a minimal spanning tree of a point cloud to a graph with cycles. We generalize this skeleton to higher dimensions and prove its optimality among all complexes that preserve topological features of data at any scale.

This is joint work with V. Kurlin and D. Lesnik.