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Computing Betti tables and minimal presentations of zero-dimensional persistent homology

Yuan Luo, Dmitriy Morozov, Luis Scoccola

TL;DR

This work addresses scalable computation of Betti tables and minimal presentations for zero-dimensional persistent homology in multiparameter (notably two-parameter) settings. By introducing minimal filtered graphs and dynamic dendrograms, the authors achieve $O(|G|)$ time for $eta_0$, $O(|G|^2)$ for minimal presentations, and $O(|G|\, ext{log}|G|)$ for full Betti tables in the $ ext{R}^2$-case, while uncovering field-independence for the first two Betti tables and field-dependent higher Betti tables. They establish a connection between algebraic invariants and connectivity properties, and provide robust algorithms for handling multi-critical filtrations with a tractable preprocessing step. An implementation in C++ with Python bindings demonstrates practical speedups and memory efficiency over existing tools on both synthetic and real datasets, enabling scalable analysis of large topological datasets. The work contributes both theoretical insights into the structure of zero-dimensional multiparameter persistence and practical methods for efficient computation and visualization of Betti descriptors.

Abstract

The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes changes. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of zero-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of zero-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

Computing Betti tables and minimal presentations of zero-dimensional persistent homology

TL;DR

This work addresses scalable computation of Betti tables and minimal presentations for zero-dimensional persistent homology in multiparameter (notably two-parameter) settings. By introducing minimal filtered graphs and dynamic dendrograms, the authors achieve time for , for minimal presentations, and for full Betti tables in the -case, while uncovering field-independence for the first two Betti tables and field-dependent higher Betti tables. They establish a connection between algebraic invariants and connectivity properties, and provide robust algorithms for handling multi-critical filtrations with a tractable preprocessing step. An implementation in C++ with Python bindings demonstrates practical speedups and memory efficiency over existing tools on both synthetic and real datasets, enabling scalable analysis of large topological datasets. The work contributes both theoretical insights into the structure of zero-dimensional multiparameter persistence and practical methods for efficient computation and visualization of Betti descriptors.

Abstract

The Betti tables of a multigraded module encode the grades at which there is an algebraic change in the module. Multigraded modules show up in many areas of pure and applied mathematics, and in particular in topological data analysis, where they are known as persistence modules, and where their Betti tables describe the places at which the homology of filtered simplicial complexes changes. Although Betti tables of singly and bigraded modules are already being used in applications of topological data analysis, their computation in the bigraded case (which relies on an algorithm that is cubic in the size of the filtered simplicial complex) is a bottleneck when working with large datasets. We show that, in the special case of zero-dimensional homology (relevant for clustering and graph classification) Betti tables of bigraded modules can be computed in log-linear time. We also consider the problem of computing minimal presentations, and show that minimal presentations of zero-dimensional persistent homology can be computed in quadratic time, regardless of the grading poset.

Paper Structure

This paper contains 33 sections, 32 theorems, 14 equations, 7 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Betti tables and persistence
  3. Two-parameter persistent homology
  4. Zero-dimensional persistent homology
  5. Contributions
  6. Structure of the paper and summary of the approach
  7. Remark about multi-critical filtrations
  8. Related work
  9. Background
  10. Graphs and filtered graphs
  11. Persistence modules and persistent sets
  12. Persistent homology and connected components of filtered graphs
  13. Projective persistence modules
  14. Resolutions, presentations, and Betti tables
  15. Theory
  16. ...and 18 more sections

Key Result

Theorem A

Let $\mathscr{P}$ be any poset, and let $(G,f)$ be a finite $\mathscr{P}$-filtered graph. algorithm:vertex-reduction computes the $0$th Betti table of $H_0(G,f; \mathbb{k})$ in $O(\,|G| \,)$ time. algorithm:minimal-presentation computes a minimal presentation (and hence the $0$th and $1$st Betti tab

Figures (7)

  • Figure 1: Left. A bifiltered graph $(G,f)$ with vertex set $\{u,v,w,x_1,x_2,x_3\}$ and edge set $\{e_1, e_2, e_3, h_1, h_2, d_1, d_2, d_3\}$. Right. The bifiltered graph schematically mapped to $\mathbb{R}^2$, together with the Betti tables $\beta_0(H_0(G,f))$ (circles), $\beta_1(H_0(G,f))$ (crosses), $\beta_2(H_0(G,f))$ (stars), and $\beta_0(H_1(G,f))$ (squares).
  • Figure 2: Left. A graph with labeled vertices and edges. Right. An $\mathbb{R}^4$-filtration of the graph on the left depicted as the Hasse diagram of the subposet of $\mathbb{R}^4$ spanned by grades at which a vertex or an edge appears.
  • Figure 3: The 3D Blob clusters of size 400. The purple and yellow clusters are closer to each other than the green one.
  • Figure 4: The three persistence modules in a minimal resolution of $H_0(G,f)$ for the bifiltered graph in \ref{['fig:graded-graph']}.
  • Figure 5: Hilbert matrix of the bifiltered graph in \ref{['fig:graded-graph']}.
  • ...and 2 more figures

Theorems & Definitions (66)

  • Theorem A
  • Theorem B
  • Theorem C
  • Theorem D
  • Lemma 2.1
  • Definition 2.3
  • Lemma 2.4
  • proof
  • Lemma 2.5
  • Definition 2.6
  • ...and 56 more