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 injective. We therefore propose that, in many cases, the collection of persistence diagrams of many small subsets of X is a better invariant. This invariant, which we call "distributed persistence," is perfectly parallelizable, more stable to outliers, and has a rich inverse theory. The map from the space of metric spaces (with the quasiisometry distance) to the space of distributed persistence invariants (with the HausdorffBottleneck distance) is globally biLipschitz. This is a much stronger property than simply being injective, as it implies that the inverse image of a small neighborhood is a small neighborhood, and is to our knowledge the only result of its kind in the TDA literature. Moreover, the inverse Lipschitz constant depends 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 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 simple covering properties. These theoretical results are complemented by synthetic experiments demonstrating the use of distributed persistence in practice.
BibTeX  Entry
@InProceedings{solomon_et_al:LIPIcs.SoCG.2022.61,
author = {Solomon, Elchanan and Wagner, Alexander and Bendich, Paul},
title = {{From Geometry to Topology: Inverse Theorems for Distributed Persistence}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {61:161:16},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959772273},
ISSN = {18688969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16069},
URN = {urn:nbn:de:0030drops160690},
doi = {10.4230/LIPIcs.SoCG.2022.61},
annote = {Keywords: Applied Topology, Persistent Homology, Inverse Problems, Subsampling}
}
Keywords: 

Applied Topology, Persistent Homology, Inverse Problems, Subsampling 
Collection: 

38th International Symposium on Computational Geometry (SoCG 2022) 
Issue Date: 

2022 
Date of publication: 

01.06.2022 
Supplementary Material: 

Software (Source Code): https://github.com/aywagner/DIPOLE 