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Efficient Algorithms for Complexes of Persistence Modules with Applications

Tamal K. Dey, Florian Russold, Shreyas N. Samaga

TL;DR

The persistence algorithm is extended as an algorithm computing the homology of a complex of free persistence or graded modules to complexes of modules that are not free, and an efficient algorithm is developed to compute the homology of a complex of presentations.

Abstract

We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.

Efficient Algorithms for Complexes of Persistence Modules with Applications

TL;DR

The persistence algorithm is extended as an algorithm computing the homology of a complex of free persistence or graded modules to complexes of modules that are not free, and an efficient algorithm is developed to compute the homology of a complex of presentations.

Abstract

We extend the persistence algorithm, viewed as an algorithm computing the homology of a complex of free persistence or graded modules, to complexes of modules that are not free. We replace persistence modules by their presentations and develop an efficient algorithm to compute the homology of a complex of presentations. To deal with inputs that are not given in terms of presentations, we give an efficient algorithm to compute a presentation of a morphism of persistence modules. This allows us to compute persistent (co)homology of instances giving rise to complexes of non-free modules. Our methods lead to a new efficient algorithm for computing the persistent homology of simplicial towers and they enable efficient algorithms to compute the persistent homology of cosheaves over simplicial towers and cohomology of persistent sheaves on simplicial complexes. We also show that we can compute the cohomology of persistent sheaves over arbitrary finite posets by reducing the computation to a computation over simplicial complexes.
Paper Structure (21 sections, 27 theorems, 39 equations, 6 figures, 1 table)

This paper contains 21 sections, 27 theorems, 39 equations, 6 figures, 1 table.

Table of Contents

  1. Introduction
  2. Persistence modules, graded modules, and presentations
  3. Computing homology of complexes of presentations
  4. Computing presentations of morphisms of persistence modules
  5. Complexes of presentations
  6. Applications
  7. Persistent homology of simplicial towers
  8. Persistent cosheaf homology over simplicial towers
  9. Cohomology of persistent sheaves over simplicial complexes
  10. Cohomology of persistent sheaves over posets
  11. Conclusion
  12. Sheaves and their persistent version
  13. Cosheaves over persistent spaces
  14. Proofs in Section \ref{['sec_presentation_algorithm']}
  15. Proofs in Section \ref{['sec_comp_presentations']}
  16. ...and 6 more sections

Key Result

Proposition 2

$\mathbf{pMod} \cong \mathbf{grMod}_{\Bbbk[t]}$.

Figures (6)

  • Figure 1: The pullback $f^*F$ of a cosheaf $F$ along an elementary collapse $f:\text{v}_3\rightarrow \text{v}_2$.
  • Figure 2: A sheaf morphism $\phi\colon F\rightarrow G$ over a triangle and its faces.
  • Figure 3: Two equivalent viewpoints of a persistent sheaf over a triangle and its faces.
  • Figure 4: A persistent sheaf $\Vec{F}$ and the corresponding sheaf of graded $\Bbbk[t]$-modules $\mathcal{M}\Vec{F}$.
  • Figure 5: Plot of run time of the algorithm to compute persistent sheaf cohomology versus number of cores being used for computation.
  • ...and 1 more figures

Theorems & Definitions (43)

  • Example 1
  • Proposition 2
  • Definition 3: Presentation of module
  • Definition 4: Presentation of morphism
  • Example 5
  • Example 6
  • Example 7
  • Proposition 8
  • Proposition 9
  • Theorem 10
  • ...and 33 more