Persistence module

A persistence module is a mathematical structure in persistent homology and topological data analysis that formally captures the persistence of topological features of an object across a range of scale parameters. A persistence module often consists of a collection of homology groups (or vector spaces if using field coefficients) corresponding to a filtration of topological spaces, and a collection of linear maps induced by the inclusions of the filtration. The concept of a persistence module was first introduced in 2005 as an application of graded modules over polynomial rings, thus importing well-developed algebraic ideas from classical commutative algebra theory to the setting of persistent homology.[1] Since then, persistence modules have been one of the primary algebraic structures studied in the field of applied topology.[2][3][4][5][6][7]

  1. ^ Zomorodian, Afra; Carlsson, Gunnar (2005). "Computing Persistent Homology". Discrete & Computational Geometry. 33 (2): 249–274. doi:10.1007/s00454-004-1146-y. ISSN 0179-5376.
  2. ^ The structure and stability of persistence modules. Frédéric Chazal, Vin De Silva, Marc Glisse, Steve Y. Oudot. Switzerland. 2016. ISBN 978-3-319-42545-0. OCLC 960458101.{{cite book}}: CS1 maint: location missing publisher (link) CS1 maint: others (link)
  3. ^ Oudot, Steve Y. (2015). Persistence theory : from quiver representations to data analysis. Providence, Rhode Island. ISBN 978-1-4704-2545-6. OCLC 918149730.{{cite book}}: CS1 maint: location missing publisher (link)
  4. ^ Polterovich, Leonid (2020). Topological persistence in geometry and analysis. Daniel Rosen, Karina Samvelyan, Jun Zhang. Providence, Rhode Island. ISBN 978-1-4704-5495-1. OCLC 1142009348.{{cite book}}: CS1 maint: location missing publisher (link)
  5. ^ Schenck, Hal (2022). Algebraic foundations for applied topology and data analysis. Cham. ISBN 978-3-031-06664-1. OCLC 1351750760.{{cite book}}: CS1 maint: location missing publisher (link)
  6. ^ Dey, Tamal K. (2022). Computational topology for data analysis. Yusu Wang. Cambridge, United Kingdom. ISBN 978-1-009-09995-0. OCLC 1281786176.{{cite book}}: CS1 maint: location missing publisher (link)
  7. ^ Rabadan, Raul; Blumberg, Andrew J. (2019). Topological Data Analysis for Genomics and Evolution: Topology in Biology. Cambridge: Cambridge University Press. doi:10.1017/9781316671665. ISBN 978-1-107-15954-9. S2CID 242498045.