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Creative Commons Attribution 4.0 International license (CC BY 4.0)
When quoting this document, please refer to the following
DOI: 10.4230/LIPIcs.SoCG.2022.58
URN: urn:nbn:de:0030-drops-160664
URL: http://dagstuhl.sunsite.rwth-aachen.de/volltexte/2022/16066/
Rolle, Alexander
The Degree-Rips Complexes of an Annulus with Outliers
Abstract
The degree-Rips bifiltration is the most computable of the parameter-free, density-sensitive bifiltrations in topological data analysis. It is known that this construction is stable to small perturbations of the input data, but its robustness to outliers is not well understood. In recent work, Blumberg-Lesnick prove a result in this direction using the Prokhorov distance and homotopy interleavings. Based on experimental evaluation, they argue that a more refined approach is desirable, and suggest the framework of homology inference. Motivated by these experiments, we consider a probability measure that is uniform with high density on an annulus, and uniform with low density on the disc inside the annulus. We compute the degree-Rips complexes of this probability space up to homotopy type, using the Adamaszek-Adams computation of the Vietoris-Rips complexes of the circle. These degree-Rips complexes are the limit objects for the Blumberg-Lesnick experiments. We argue that the homology inference approach has strong explanatory power in this case, and suggest studying the limit objects directly as a strategy for further work.
BibTeX - Entry
@InProceedings{rolle:LIPIcs.SoCG.2022.58,
author = {Rolle, Alexander},
title = {{The Degree-Rips Complexes of an Annulus with Outliers}},
booktitle = {38th International Symposium on Computational Geometry (SoCG 2022)},
pages = {58:1--58:14},
series = {Leibniz International Proceedings in Informatics (LIPIcs)},
ISBN = {978-3-95977-227-3},
ISSN = {1868-8969},
year = {2022},
volume = {224},
editor = {Goaoc, Xavier and Kerber, Michael},
publisher = {Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
address = {Dagstuhl, Germany},
URL = {https://drops.dagstuhl.de/opus/volltexte/2022/16066},
URN = {urn:nbn:de:0030-drops-160664},
doi = {10.4230/LIPIcs.SoCG.2022.58},
annote = {Keywords: multi-parameter persistent homology, stability, homology inference}
}
