Stability for Inference with Persistent Homology Rank Functions

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dc.contributor.authorWang, Qiquanen_US
dc.contributor.authorGarcía-Redondo, Inésen_US
dc.contributor.authorFaugère, Pierreen_US
dc.contributor.authorHenselman-Petrusek, Gregoryen_US
dc.contributor.authorMonod, Antheaen_US
dc.contributor.editorHu, Ruizhenen_US
dc.contributor.editorLefebvre, Sylvainen_US
dc.date.accessioned2024-06-20T07:55:23Z
dc.date.available2024-06-20T07:55:23Z
dc.date.issued2024
dc.description.abstractPersistent homology barcodes and diagrams are a cornerstone of topological data analysis that capture the ''shape'' of a wide range of complex data structures, such as point clouds, networks, and functions. However, their use in statistical settings is challenging due to their complex geometric structure. In this paper, we revisit the persistent homology rank function, which is mathematically equivalent to a barcode and persistence diagram, as a tool for statistics and machine learning. Rank functions, being functions, enable the direct application of the statistical theory of functional data analysis (FDA)-a domain of statistics adapted for data in the form of functions. A key challenge they present over barcodes in practice, however, is their lack of stability-a property that is crucial to validate their use as a faithful representation of the data and therefore a viable summary statistic. In this paper, we fill this gap by deriving two stability results for persistent homology rank functions under a suitable metric for FDA integration. We then study the performance of rank functions in functional inferential statistics and machine learning on real data applications, in both single and multiparameter persistent homology. We find that the use of persistent homology captured by rank functions offers a clear improvement over existing non-persistence-based approaches.en_US
dc.description.number5
dc.description.sectionheadersShape Analysis
dc.description.seriesinformationComputer Graphics Forum
dc.description.volume43
dc.identifier.doi10.1111/cgf.15142
dc.identifier.issn1467-8659
dc.identifier.pages18 pages
dc.identifier.urihttps://doi.org/10.1111/cgf.15142
dc.identifier.urihttps://diglib.eg.org/handle/10.1111/cgf15142
dc.publisherThe Eurographics Association and John Wiley & Sons Ltd.en_US
dc.rightsAttribution 4.0 International License
dc.rights.urihttps://creativecommons.org/licenses/by/4.0/
dc.subjectCCS Concepts: Mathematics of computing → Algebraic topology; Theory of computation → Computational geometry; General and reference → Performance
dc.subjectMathematics of computing → Algebraic topology
dc.subjectTheory of computation → Computational geometry
dc.subjectGeneral and reference → Performance
dc.titleStability for Inference with Persistent Homology Rank Functionsen_US
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