Numeric invariants from multidimensional persistence
Abstract
Topological data analysis is the study of data using techniques from algebraic topology. Often, one begins with a finite set of points representing data and a “filter” function which assigns a real number to each datum. Using both the data and the filter function, one can construct a filtered complex for further analysis. For example, applying the homology functor to the filtered complex produces an algebraic object known as a “onedimensional persistence module”, which can often be interpreted as a finite set of intervals representing various geometric features in the data. If one runs the above process incorporating multiple filter functions simultaneously, one instead obtains a multidimensional persistence module. Unfortunately, these are much more difficult to interpret. In this article, we analyze the space of multidimensional persistence modules from the perspective of algebraic geometry. First we build a moduli space of a certain subclass of easily analyzed multidimensional persistence modules, which we construct specifically to capture much of the information which can be gained by using multidimensional persistence instead of onedimensional persistence. Fruthermore, we argue that the global sections of this space provide interesting numeric invariants when evaluated against our subclass of multidimensional persistence modules. Finally, we extend these globalmore »
 Authors:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Stanford Univ., CA (United States)
 Publication Date:
 Research Org.:
 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Sponsoring Org.:
 USDOE National Nuclear Security Administration (NNSA)
 OSTI Identifier:
 1361045
 Report Number(s):
 SAND20174533J
Journal ID: ISSN 23671726; PII: 3
 Grant/Contract Number:
 AC0494AL85000
 Resource Type:
 Journal Article: Accepted Manuscript
 Journal Name:
 Journal of Applied and Computational Topology
 Additional Journal Information:
 Journal Volume: 1; Journal Issue: 1; Journal ID: ISSN 23671726
 Publisher:
 Springer
 Country of Publication:
 United States
 Language:
 English
 Subject:
 58 GEOSCIENCES; 97 MATHEMATICS AND COMPUTING; multidimensional persistent homology; numeric invariants; applied topology
Citation Formats
Skryzalin, Jacek, and Carlsson, Gunnar. Numeric invariants from multidimensional persistence. United States: N. p., 2017.
Web. doi:10.1007/s414680170003z.
Skryzalin, Jacek, & Carlsson, Gunnar. Numeric invariants from multidimensional persistence. United States. doi:10.1007/s414680170003z.
Skryzalin, Jacek, and Carlsson, Gunnar. Fri .
"Numeric invariants from multidimensional persistence". United States.
doi:10.1007/s414680170003z. https://www.osti.gov/servlets/purl/1361045.
@article{osti_1361045,
title = {Numeric invariants from multidimensional persistence},
author = {Skryzalin, Jacek and Carlsson, Gunnar},
abstractNote = {Topological data analysis is the study of data using techniques from algebraic topology. Often, one begins with a finite set of points representing data and a “filter” function which assigns a real number to each datum. Using both the data and the filter function, one can construct a filtered complex for further analysis. For example, applying the homology functor to the filtered complex produces an algebraic object known as a “onedimensional persistence module”, which can often be interpreted as a finite set of intervals representing various geometric features in the data. If one runs the above process incorporating multiple filter functions simultaneously, one instead obtains a multidimensional persistence module. Unfortunately, these are much more difficult to interpret. In this article, we analyze the space of multidimensional persistence modules from the perspective of algebraic geometry. First we build a moduli space of a certain subclass of easily analyzed multidimensional persistence modules, which we construct specifically to capture much of the information which can be gained by using multidimensional persistence instead of onedimensional persistence. Fruthermore, we argue that the global sections of this space provide interesting numeric invariants when evaluated against our subclass of multidimensional persistence modules. Finally, we extend these global sections to the space of all multidimensional persistence modules and discuss how the resulting numeric invariants might be used to study data. This paper extends the results of Adcock et al. (Homol Homotopy Appl 18(1), 381–402, 2016) by constructing numeric invariants from the computation of a multidimensional persistence module as given by Carlsson et al. (J Comput Geom 1(1), 72–100, 2010).},
doi = {10.1007/s414680170003z},
journal = {Journal of Applied and Computational Topology},
number = 1,
volume = 1,
place = {United States},
year = {Fri May 19 00:00:00 EDT 2017},
month = {Fri May 19 00:00:00 EDT 2017}
}

In this paper, we analyze the space of multidimensional persistence modules from the perspectives of algebraic geometry. We first build a moduli space of a certain subclass of easily analyzed multidimensional persistence modules, which we construct specifically to capture much of the information which can be gained by using multidimensional persistence over onedimensional persistence. We argue that the global sections of this space provide interesting numeric invariants when evaluated against our subclass of multidimensional persistence modules. Lastly, we extend these global sections to the space of all multidimensional persistence modules and discuss how the resulting numeric invariants might be usedmore »

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