Stratifying Multiparameter Persistent Homology
Abstract
A fundamental tool in topological data analysis is persistent homology, which allows extraction of information from complex datasets in a robust way. Persistent homology assigns a module over a principal ideal domain to a one-parameter family of spaces obtained from the data. In applications, data often depend on several parameters, and in this case one is interested in studying the persistent homology of a multiparameter family of spaces associated to the data. While the theory of persistent homology for one-parameter families is well understood, the situation for multiparameter families is more delicate. Following Carlsson and Zomorodian, we recast the problem in the setting of multigraded algebra, and we propose multigraded Hilbert series, multigraded associated primes, and local cohomology as invariants for studying multiparameter persistent homology. Multigraded associated primes provide a stratification of the region where a multigraded module does not vanish, while multigraded Hilbert series and local cohomology give a measure of the size of components of the module supported on different strata. These invariants generalize in a suitable sense the invariant for the one-parameter case.
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Information & Authors
Information
Published In
SIAM Journal on Applied Algebra and Geometry
Pages: 439 - 471
DOI: 10.1137/18M1224350
ISSN (online): 2470-6566
Copyright
© 2019, Society for Industrial and Applied Mathematics.
History
Submitted: 2 November 2018
Accepted: 3 June 2019
Published online: 5 September 2019
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Funding Information
Emirates Foundation https://doi.org/10.13039/501100005930
National Science Foundation https://doi.org/10.13039/100000001 : 1818646
Engineering and Physical Sciences Research Council https://doi.org/10.13039/501100000266 : EP/N510129/1, EP/K041096/1, EP/R018472-1, EP/P511377/1
Royal Society https://doi.org/10.13039/501100000288
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