Abstract
Given a persistence diagram with n points, we give an algorithm that produces a sequence of n persistence diagrams converging in bottleneck distance to the input diagram, the ith of which has i distinct (weighted) points and is a 2approximation to the closest persistence diagram with that many distinct points. For each approximation, we precompute the optimal matching between the ith and the (i+1)st. Perhaps surprisingly, the entire sequence of diagrams as well as the sequence of matchings can be represented in O(n) space. The main approach is to use a variation of the greedy permutation of the persistence diagram to give good Hausdorff approximations and assign weights to these subsets. We give a new algorithm to efficiently compute this permutation, despite the high implicit dimension of points in a persistence diagram due to the effect of the diagonal. The sketches are also structured to permit fast (linear time) approximations to the Hausdorff distance between diagrams  a lower bound on the bottleneck distance. For approximating the bottleneck distance, sketches can also be used to compute a linearsize neighborhood graph directly, obviating the need for geometric data structures used in stateoftheart methods for bottleneck computation.
BibTeX  Entry
@InProceedings{sheehy_et_al:LIPIcs.SoCG.2021.57,
author = {Sheehy, Donald R. and Sheth, Siddharth},
title = {{Sketching Persistence Diagrams}},
booktitle = {37th International Symposium on Computational Geometry (SoCG 2021)},
pages = {57:157:15},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {9783959771849},
ISSN = {18688969},
year = {2021},
volume = {189},
editor = {Buchin, Kevin and Colin de Verdi\`{e}re, \'{E}ric},
publisher = {Schloss Dagstuhl  LeibnizZentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2021/13856},
URN = {urn:nbn:de:0030drops138569},
doi = {10.4230/LIPIcs.SoCG.2021.57},
annote = {Keywords: Bottleneck Distance, Persistent Homology, Approximate Persistence Diagrams}
}
Keywords: 

Bottleneck Distance, Persistent Homology, Approximate Persistence Diagrams 
Collection: 

37th International Symposium on Computational Geometry (SoCG 2021) 
Issue Date: 

2021 
Date of publication: 

02.06.2021 