A New Formulation of 1-D Persistent Homology Via Correspondence Modules

Released on by Haibin Hang
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  • A New Formulation of 1-D Persistent Homology Via Correspondence Modules Book Detail

  • Author : Haibin Hang
  • Release Date : 2020
  • Publisher :
  • Genre : Mathematics
  • Pages : 0
  • ISBN 13 :
  • File Size : 66,66 MB
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A New Formulation of 1-D Persistent Homology Via Correspondence Modules by Haibin Hang PDF Summary

Book Description: We develop a unifying framework for the treatment of various types of persistent homology using the notion of correspondence modules. In this formulation, morphisms between vector spaces are partial linear relations, as opposed to linear mappings. In the one-dimensional case, among other things, this allows us to: (i) treat persistence modules and zigzag modules as algebraic objects of the same type; (ii) give a categorical formulation of zigzag structures that allows a continuous parameter; and (iii) construct barcodes associated with a filtered space that are richer in geometric information. A structural analysis of 1D persistence modules is carried out at the level of sections of correspondence modules that yield sheaf-like structures, termed persistence sheaves. Under some tameness hypotheses, we prove interval decomposition theorems for persistence sheaves and correspondence modules, as well as an isometry theorem for persistence diagrams obtained from interval decompositions of persistence sheaves. Applications of correspondence modules in levelset persistence and $1$D slicing of $2$D persistence modules are introduced. Also, a Mayer-Vietoris sequence in correspondence module category is formalized.

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