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Computing Representatives of Persistent Homology Generators with a Double Twist

Tuyen Pham, Hubert Wagner

TL;DR

The twist algorithm of Chen and Kerber is extended, based on a new technique called saving, which supplements their existing killing technique and improves the efficiency of computing representatives of persistent homology generators.

Abstract

With the growing availability of efficient tools, persistent homology is becoming a useful methodology in a variety of applications. Significant work has been devoted to implementing tools for persistent homology diagrams; however, computing representative cycles corresponding to each point in the diagram can still be inefficient. To circumvent this problem, we extend the twist algorithm of Chen and Kerber. Our extension is based on a new technique we call saving, which supplements their existing killing technique. The resulting two-pass strategy can be realized using an existing matrix reduction implementation as a black-box and improves the efficiency of computing representatives of persistent homology generators. We prove the correctness of the new approach and experimentally show its performance.

Computing Representatives of Persistent Homology Generators with a Double Twist

TL;DR

The twist algorithm of Chen and Kerber is extended, based on a new technique called saving, which supplements their existing killing technique and improves the efficiency of computing representatives of persistent homology generators.

Abstract

With the growing availability of efficient tools, persistent homology is becoming a useful methodology in a variety of applications. Significant work has been devoted to implementing tools for persistent homology diagrams; however, computing representative cycles corresponding to each point in the diagram can still be inefficient. To circumvent this problem, we extend the twist algorithm of Chen and Kerber. Our extension is based on a new technique we call saving, which supplements their existing killing technique. The resulting two-pass strategy can be realized using an existing matrix reduction implementation as a black-box and improves the efficiency of computing representatives of persistent homology generators. We prove the correctness of the new approach and experimentally show its performance.
Paper Structure (7 sections, 1 theorem, 3 figures, 1 table, 3 algorithms)

This paper contains 7 sections, 1 theorem, 3 figures, 1 table, 3 algorithms.

Table of Contents

  1. Overview
  2. Mathematical background
  3. Related work
  4. Existing algorithms
  5. Double twist strategy
  6. Experiments
  7. Summary

Key Result

Proposition 1

The output of the double twist algorithm applied to a filtration $F$ coincides with the reduced matrix $B'$ output by the twist algorithm applied to the boundary matrix $B$ of $F$.

Figures (3)

  • Figure 1: Example of a Vietoris--Rips filtration which approximates the topology of the growing union of disks. Typically information about the birth and death of topological features would be encoded as a persistence diagram. The cycle highlighted in yellow (born at radius 0.71, and destroyed at 1.06) is one representative cycle returned by the algorithm we propose.
  • Figure 2: Standard boundary matrix reduction of a filtration of the 1-skeleton of a tetrahedron. The order in which the vertices and edges are added is determined by the numbers. Intermediate state of each column during reduction is shown.
  • Figure 3: Double twist applied to the same filtration as in Figure \ref{['fig:boundaryReduc']}. After the first pass (top), simplices $4,5,6$ are saved as indicated by the array (middle) representing the saved-simplices variable. After the second pass (bottom) we obtain a matrix identical to the reduced matrix in Figure \ref{['fig:boundaryReduc']}.

Theorems & Definitions (1)

  • Proposition 1: Saving Works