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Keeping it sparse: Computing Persistent Homology revisited

Ulrich Bauer, Talha Bin Masood, Barbara Giunti, Guillaume Houry, Michael Kerber, Abhishek Rathod

TL;DR

This work proposes two novel variants of the standard algorithm, called swap and retrospective reductions, and presents novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction.

Abstract

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.

Keeping it sparse: Computing Persistent Homology revisited

TL;DR

This work proposes two novel variants of the standard algorithm, called swap and retrospective reductions, and presents novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction.

Abstract

In this work, we study several variants of matrix reduction via Gaussian elimination that try to keep the reduced matrix sparse. The motivation comes from the growing field of topological data analysis where matrix reduction is the major subroutine to compute barcodes, the main invariant therein. We propose two novel variants of the standard algorithm, called swap and retrospective reductions. We test them on a large collection of data against other known variants to compare their efficiency, and we find that sometimes they provide a considerable speed-up. We also present novel output-sensitive bounds for the retrospective variant which better explain the discrepancy between the cubic worst-case complexity bound and the almost linear practical behavior of matrix reduction. Finally, we provide several constructions on which one of the variants performs strictly better than the others.
Paper Structure (27 sections, 9 theorems, 5 equations, 5 figures, 6 tables)

This paper contains 27 sections, 9 theorems, 5 equations, 5 figures, 6 tables.

Table of Contents

  1. Introduction
  2. Motivation
  3. Results
  4. Related work
  5. Preliminaries
  6. Matrix reduction
  7. Filtered simplicial complex and boundary matrix
  8. Persistence pairs
  9. Clear and compress
  10. Column representations
  11. Dualization
  12. Sparsification variants
  13. Swap reduction
  14. Exhaustive reduction
  15. Retrospective reduction
  16. ...and 12 more sections

Key Result

Corollary 1

Any matrix reduction algorithm that preserves the ranks of the submatrices $D[\geq i, \leq j]$, for all $i,j\in \{1,\dots, N\}$ is a valid barcode algorithm.

Figures (5)

  • Figure 1: Depiction of $K_1$ and $K_2$.
  • Figure 2: (Sub)matrix of $K_1$
  • Figure 3: Matrix for $K_2$ (with $n=4$)
  • Figure 4: Matrix for $K_3$ (with $n=4$)
  • Figure 5: Matrix for $K_4$ (with $n=4$)

Theorems & Definitions (9)

  • Corollary 1
  • Proposition 2
  • Lemma 5
  • Lemma 6
  • Lemma 7
  • Corollary 8
  • Proposition 9
  • Proposition 10
  • Proposition 11