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Fast computation of persistent homology representatives with involuted persistent homology

Matija Čufar, Žiga Virk

Abstract

Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are geometric manifestations of the corresponding holes and often carry desirable information. We propose a new method of extraction of persistent homology representatives using cohomology. In a nutshell, we first compute persistent cohomology and use the obtained information to significantly improve the running time of the direct persistent homology computations. This algorithm applied to Rips filtrations generally computes persistent homology representatives much faster than the standard methods.

Fast computation of persistent homology representatives with involuted persistent homology

Abstract

Persistent homology is typically computed through persistent cohomology. While this generally improves the running time significantly, it does not facilitate extraction of homology representatives. The mentioned representatives are geometric manifestations of the corresponding holes and often carry desirable information. We propose a new method of extraction of persistent homology representatives using cohomology. In a nutshell, we first compute persistent cohomology and use the obtained information to significantly improve the running time of the direct persistent homology computations. This algorithm applied to Rips filtrations generally computes persistent homology representatives much faster than the standard methods.

Paper Structure

This paper contains 16 sections, 2 theorems, 12 equations, 4 figures, 2 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Theoretical background
  3. Persistent homology
  4. Reduction algorithm for persistent homology
  5. Representatives
  6. Persistent cohomology
  7. Rips complexes
  8. Algorithm
  9. The algorithm
  10. Details on the algorithm
  11. Example
  12. Experiments
  13. Analysis
  14. Comparisons for coboundary, boundary and involuted boundary matrix reductions.
  15. Comparison of computation times between cohomology, involuted homology, and Eirene.
  16. ...and 1 more sections

Key Result

Theorem 1

Let $K \hookrightarrow L$ be an inclusion of simplicial complexes. Then for each dimension $p$ there exists a commutative diagram \xymatrix{H_p(K) \ar[d] \ar[r] & H_p(L) \ar[d] \\ H^p(K) & H^p(L) \ar[l]}with the vertical maps being isomorphisms.

Figures (4)

  • Figure 1: Comparison of persistence cocycles and persistence cycles on a data set consisting of 20 points sampled randomly from a circle (red), 12 points sampled uniformly from a circle (purple), and 40 points sampled randomly from an annulus (green).
  • Figure 2: Timings of our code and Eirene with increasing maximum homology dimension on a data set of 50 random points in $\mathbb{R}^{16}$. The left pane shows relative speedups compared to the homology computation, the right panes show the elapsed time in seconds in a logarithmic scale.
  • Figure 3: Timings of computing one-dimensional persistent homology with our code and Eirene on increasing number of points $N$ in three data sets. The left panes show relative speedups compared to the homology computation, the right panes show the elapsed time in seconds in a logarithmic scale. The maximal computeds dimension of persistent homology are noted in Table \ref{['tab:eirene']}. The inconsistent improvements in the dragon1000 and hiv can be explained by the fact that adding points to a complex data set may drastically change its persistent homological features.
  • Figure 4: Speedups compared to homology computations for various data sets.

Theorems & Definitions (4)

  • Theorem 1
  • Remark 1
  • Proposition 1
  • proof