Nothing Special   »   [go: up one dir, main page]
Document Open Access Logo

Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension

Authors Mickaël Buchet, Emerson G. Escolar

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2018.15.pdf
  • Filesize: 490 kB
  • 13 pages

Document Identifiers

Author Details

Mickaël Buchet
Emerson G. Escolar

Cite As Get BibTex

Mickaël Buchet and Emerson G. Escolar. Realizations of Indecomposable Persistence Modules of Arbitrarily Large Dimension. In 34th International Symposium on Computational Geometry (SoCG 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 99, pp. 15:1-15:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018) https://doi.org/10.4230/LIPIcs.SoCG.2018.15

Abstract

While persistent homology has taken strides towards becoming a widespread tool for data analysis, multidimensional persistence has proven more difficult to apply. One reason is the serious drawback of no longer having a concise and complete descriptor analogous to the persistence diagrams of the former. We propose a simple algebraic construction to illustrate the existence of infinite families of indecomposable persistence modules over regular grids of sufficient size. On top of providing a constructive proof of representation infinite type, we also provide realizations by topological spaces and Vietoris-Rips filtrations, showing that they can actually appear in real data and are not the product of degeneracies.

Subject Classification

Keywords
  • persistent homology
  • multi-persistence
  • representation theory
  • quivers
  • commutative ladders
  • Vietoris-Rips filtration

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

References

  1. Hideto Asashiba, Emerson G. Escolar, Yasuaki Hiraoka, and Hiroshi Takeuchi. Matrix method for persistence modules on commutative ladders of finite type. arXiv preprint arXiv:1706.10027, 2017. Google Scholar
  2. Ibrahim Assem, Andrzej Skowronski, and Daniel Simson. Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory, volume 65. Cambridge University Press, 2006. Google Scholar
  3. Michael Barot. Introduction to the representation theory of algebras. Springer, 2014. Google Scholar
  4. Gunnar Carlsson and Vin de Silva. Zigzag persistence. Foundations of computational mathematics, 10(4):367-405, 2010. Google Scholar
  5. Gunnar Carlsson and Afra Zomorodian. The theory of multidimensional persistence. Discrete &Computational Geometry, 42(1):71-93, 2009. Google Scholar
  6. Lorin Crawford, Anthea Monod, Andrew X. Chen, Sayan Mukherjee, and Raúl Rabadán. Topological summaries of tumor images improve prediction of disease free survival in glioblastoma multiforme. arXiv preprint arXiv:1611.06818, 2016. Google Scholar
  7. Mary-Lee Dequeant, Sebastian Ahnert, Herbert Edelsbrunner, Thomas M. A. Fink, Earl F. Glynn, Gaye Hattem, Andrzej Kudlicki, Yuriy Mileyko, Jason Morton, Arcady R. Mushegian, et al. Comparison of pattern detection methods in microarray time series of the segmentation clock. PLoS One, 3(8):e2856, 2008. Google Scholar
  8. Herbert Edelsbrunner, David Letscher, and Afra Zomorodian. Topological persistence and simplification. Discrete Comput Geom, 28:511-533, 2002. Google Scholar
  9. Emerson G. Escolar and Yasuaki Hiraoka. Persistence modules on commutative ladders of finite type. Discrete &Computational Geometry, 55(1):100-157, 2016. Google Scholar
  10. Chad Giusti, Eva Pastalkova, Carina Curto, and Vladimir Itskov. Clique topology reveals intrinsic geometric structure in neural correlations. Proceedings of the National Academy of Sciences, 112(44):13455-13460, 2015. Google Scholar
  11. Heather A Harrington, Nina Otter, Hal Schenck, and Ulrike Tillmann. Stratifying multiparameter persistent homology. arXiv preprint arXiv:1708.07390, 2017. Google Scholar
  12. Yasuaki Hiraoka, Takenobu Nakamura, Akihiko Hirata, Emerson G. Escolar, Kaname Matsue, and Yasumasa Nishiura. Hierarchical structures of amorphous solids characterized by persistent homology. Proceedings of the National Academy of Sciences, 113(26):7035-7040, 2016. Google Scholar
  13. Lida Kanari, Paweł Dłotko, Martina Scolamiero, Ran Levi, Julian Shillcock, Kathryn Hess, and Henry Markram. A topological representation of branching neuronal morphologies. Neuroinformatics, pages 1-11, 2017. Google Scholar
  14. Yongjin Lee, Senja D. Barthel, Paweł Dłotko, S. Mohamad Moosavi, Kathryn Hess, and Berend Smit. Quantifying similarity of pore-geometry in nanoporous materials. Nature Communications, 8, 2017. Google Scholar
  15. Michael Lesnick and Matthew Wright. Interactive visualization of 2-d persistence modules. arXiv preprint arXiv:1512.00180, 2015. Google Scholar
  16. James R. Munkres. Elements of algebraic topology, volume 2. Addison-Wesley Menlo Park, 1984. Google Scholar
  17. Monica Nicolau, Arnold J. Levine, and Gunnar Carlsson. Topology based data analysis identifies a subgroup of breast cancers with a unique mutational profile and excellent survival. Proceedings of the National Academy of Sciences, 108(17):7265-7270, 2011. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact