Numeric invariants from multidimensional persistence
Abstract
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 used to study data.
 Authors:

 Sandia National Lab. (SNLNM), Albuquerque, NM (United States)
 Stanford Univ., Stanford, 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:
 1335203
 Report Number(s):
 SAND20168670J
Journal ID: ISSN 23671726; 647142
 DOE Contract Number:
 AC0494AL85000
 Resource Type:
 Journal Article
 Journal Name:
 Journal of Applied and Computational Topology
 Additional Journal Information:
 Journal Volume: 1; Journal Issue: 1; Related Information: See SAND20174533J for final version; Journal ID: ISSN 23671726
 Publisher:
 Springer
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING; multidimensional persistent homology; numeric invariants; applies 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.
@article{osti_1335203,
title = {Numeric invariants from multidimensional persistence},
author = {Skryzalin, Jacek and Carlsson, Gunnar},
abstractNote = {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 used to study data.},
doi = {10.1007/s414680170003z},
journal = {Journal of Applied and Computational Topology},
issn = {23671726},
number = 1,
volume = 1,
place = {United States},
year = {2017},
month = {5}
}
Works referenced in this record:
The ring of algebraic functions on persistence bar codes
journal, January 2016
 Adcock, Aaron; Carlsson, Erik; Carlsson, Gunnar
 Homology, Homotopy and Applications, Vol. 18, Issue 1
Topology and data
journal, January 2009
 Carlsson, Gunnar
 Bulletin of the American Mathematical Society, Vol. 46, Issue 2
Topological pattern recognition for point cloud data
journal, May 2014
 Carlsson, Gunnar
 Acta Numerica, Vol. 23
The Theory of Multidimensional Persistence
journal, April 2009
 Carlsson, Gunnar; Zomorodian, Afra
 Discrete & Computational Geometry, Vol. 42, Issue 1