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Title: 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 one-dimensional 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:
 [1];  [2]
  1. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
  2. Stanford Univ., Stanford, CA (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA)
OSTI Identifier:
1335203
Report Number(s):
SAND-2016-8670J
Journal ID: ISSN 2367-1726; 647142
DOE Contract Number:  
AC04-94AL85000
Resource Type:
Journal Article
Journal Name:
Journal of Applied and Computational Topology
Additional Journal Information:
Journal Volume: 1; Journal Issue: 1; Related Information: See SAND--2017-4533J for final version; Journal ID: ISSN 2367-1726
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/s41468-017-0003-z.
Skryzalin, Jacek, & Carlsson, Gunnar. Numeric invariants from multidimensional persistence. United States. doi:10.1007/s41468-017-0003-z.
Skryzalin, Jacek, and Carlsson, Gunnar. Fri . "Numeric invariants from multidimensional persistence". United States. doi:10.1007/s41468-017-0003-z.
@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 one-dimensional 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/s41468-017-0003-z},
journal = {Journal of Applied and Computational Topology},
issn = {2367-1726},
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
  • DOI: 10.4310/HHA.2016.v18.n1.a21

Topology and data
journal, January 2009


Topological pattern recognition for point cloud data
journal, May 2014


The Theory of Multidimensional Persistence
journal, April 2009