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Fast free resolutions of bifiltered chain complexes

Ulrich Bauer, Tamal K. Dey, Michael Kerber, Florian Russold, Matthias Söls

Abstract

In a $k$-critical bifiltration, every simplex enters along a staircase with at most $k$ steps. Examples with $k>1$ include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a $k$-critical bifiltration into a $1$-critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size. We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a $k$-critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.

Fast free resolutions of bifiltered chain complexes

Abstract

In a -critical bifiltration, every simplex enters along a staircase with at most steps. Examples with include degree-Rips bifiltrations and models of the multicover bifiltration. We consider the problem of converting a -critical bifiltration into a -critical (i.e. free) chain complex with equivalent homology. This is known as computing a free resolution of the underlying chain complex and is a first step toward post-processing such bifiltrations. We present two algorithms. The first one computes free resolutions corresponding to path graphs and assembles them to a chain complex by computing additional maps. The simple combinatorial structure of path graphs leads to good performance in practice, as demonstrated by extensive experiments. However, its worst-case bound is quadratic in the input size because long paths might yield dense boundary matrices in the output. Our second algorithm replaces the simplex-wise path graphs with ones that maintain short paths which leads to almost linear runtime and output size. We demonstrate that pre-computing a free resolution speeds up the task of computing a minimal presentation of the homology of a -critical bifiltration in a fixed dimension. Furthermore, our findings show that a chain complex that is minimal in terms of generators can be asymptotically larger than the non-minimal output complex of our second algorithm in terms of description size.

Paper Structure

This paper contains 38 sections, 21 theorems, 19 equations, 17 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Motivation and problem statement.
  3. Contributions.
  4. Bifiltered chain complexes
  5. Bifiltrations.
  6. Bipersistence modules.
  7. Chain complexes.
  8. Data representation.
  9. Free implicit representations of homology
  10. FI-reps.
  11. The Chacholski--Scolamiero--Vaccarino algorithm.
  12. Path algorithm
  13. Description.
  14. Path algorithm.
  15. Complexity.
  16. ...and 23 more sections

Key Result

Lemma 1

For any $g_\tau^{x_j}$, $g_\tau^{x_\ell}$ in $G_{i-1}$, there is a unique set of distinct relations $r_\tau^{w_j},\ldots,r^{w_{\ell-1}}_\tau$ in $R_{i-1}$ such that $p^1_{i-1}(r_\tau^{w_j}+\cdots+r^{w_{\ell-1}}_\tau)=g_\tau^{x_j}+g_\tau^{x_\ell}$.

Figures (17)

  • Figure 1: The bifiltration on the left is $1$-critical while the bifiltration on the right is $3$-critical. The support of every simplex is an upset, as visualized for two edges (red and blue, respectively).
  • Figure 2: The upset module $U_\sigma$ induced by the simplex $\sigma$.
  • Figure 3: $f_{m+1}^0$ sends each generator $g_\sigma^x$ to a generator $g_\tau^y$ of the facet $\tau$ of $\sigma$ whose grade $y$ is in the downset of $x$ (say, with the smallest first coordinate).
  • Figure 4: The top row illustrates the free resolution of $U_\sigma$ as in Diagram \ref{['diag:free_res']} and the middle row shows its path resolution. We often visualize both at once as in the bottom row.
  • Figure 5: $f_i^1$ sends each edge $r_\sigma^y$ to the path connecting the images of its endpoints under $f_i^0$.
  • ...and 12 more figures

Theorems & Definitions (22)

  • Lemma 1: Path Lemma
  • Theorem 2
  • Proposition 3
  • Lemma 3: Log-path Lemma
  • Lemma 3
  • Lemma 3
  • Theorem 4
  • Theorem 5
  • Lemma 6
  • Proposition 6
  • ...and 12 more