eng
Schloss Dagstuhl – Leibniz-Zentrum für Informatik
Leibniz International Proceedings in Informatics
1868-8969
2021-06-02
57:1
57:15
10.4230/LIPIcs.SoCG.2021.57
article
Sketching Persistence Diagrams
Sheehy, Donald R.
1
https://orcid.org/0000-0002-9177-2713
Sheth, Siddharth
1
North Carolina State University, Raleigh, NC, USA
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 2-approximation 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 linear-size neighborhood graph directly, obviating the need for geometric data structures used in state-of-the-art methods for bottleneck computation.
https://drops.dagstuhl.de/storage/00lipics/lipics-vol189-socg2021/LIPIcs.SoCG.2021.57/LIPIcs.SoCG.2021.57.pdf
Bottleneck Distance
Persistent Homology
Approximate Persistence Diagrams