[Paper Review] Computing Minimal Presentations and Bigraded Betti Numbers of 2-Parameter Persistent Homology

Michael Lesnick, Matthew Wright|arXiv (Cornell University)|Feb 15, 2019

Topological and Geometric Data Analysis4 citations

TL;DR

This paper presents an efficient algorithm for computing minimal presentations and bigraded Betti numbers of 2-parameter persistent homology modules, using matrix reduction on a short chain complex of free modules. The method runs in O(|X|³ + |Y|³ + |Z|³) time and O(|X|² + |Y|² + |Z|²) memory, outperforming Singular and Macaulay2 in practice, and enables faster computation in the RIVET software for topological data analysis.

ABSTRACT

Motivated by applications to topological data analysis, we give an efficient algorithm for computing a (minimal) presentation of a bigraded $K[x,y]$-module $M$, where $K$ is a field. The algorithm takes as input a short chain complex of free modules $X\xrightarrow{f} Y \xrightarrow{g} Z$ such that $M\cong \ker{g}/\mathrm{im}{f}$. It runs in time $O(|X|^3+|Y|^3+|Z|^3)$ and requires $O(|X|^2+|Y|^2+|Z|^2)$ memory, where $|\cdot |$ denotes the rank. Given the presentation computed by our algorithm, the bigraded Betti numbers of $M$ are readily computed. Our approach is based on a simple matrix reduction algorithm, slight variants of which compute kernels of morphisms between free modules, minimal generating sets, and Gröbner bases. Our algorithm for computing minimal presentations has been implemented in RIVET, a software tool for the visualization and analysis of two-parameter persistent homology. In experiments on topological data analysis problems, our implementation outperforms the standard computational commutative algebra packages Singular and Macaulay2 by a wide margin.

Motivation & Objective

• To develop an efficient algorithm for computing minimal presentations of bigraded $K[x,y]$-modules arising in 2-parameter persistent homology.
• To enable fast and scalable computation of bigraded Betti numbers as a key invariant for topological data analysis.
• To address the computational bottleneck in tools like RIVET by optimizing the core algebraic computation pipeline.
• To outperform standard computational commutative algebra packages such as Singular and Macaulay2 on realistic TDA workloads.
• To support interactive visualization and analysis of bipersistence modules via efficient algebraic intermediates.

Proposed method

• The algorithm takes a short chain complex $X \xrightarrow{f} Y \xrightarrow{g} Z$ as input, with $M \cong \ker g / \operatorname{im} f$, and computes a minimal presentation via matrix reduction.
• It uses a variant of matrix reduction to compute kernels, minimal generating sets, and Gröbner bases over the polynomial ring $K[x,y]$.
• The method performs row and column operations on the matrices representing $f$ and $g$, reducing them to a minimal form while preserving the module structure.
• The algorithm is parallelized for minimization, significantly reducing runtime on large instances.
• It computes bigraded Betti numbers directly from the minimal presentation using standard homological algebra techniques.
• The implementation is integrated into RIVET, a tool for visualizing bipersistence modules, enabling interactive analysis.

Experimental results

Research questions

• RQ1 Can a cubic-time algorithm be designed for computing minimal presentations of 2-parameter persistence modules with optimal memory usage?
• RQ2 How does the performance of this algorithm compare to established packages like Singular and Macaulay2 in practical TDA applications?
• RQ3 Can the algorithm scale to large data sets where existing tools fail due to memory or time constraints?
• RQ4 What is the impact of parallelization on the minimization step of the presentation computation?
• RQ5 Can the algorithm be extended to handle more complex bifiltrations, such as degree-Rips or multi-critical filtrations, directly?

Key findings

• The algorithm runs in $O(|X|^3 + |Y|^3 + |Z|^3)$ time and $O(|X|^2 + |Y|^2 + |Z|^2)$ memory, with the input size defined by the ranks of the modules.
• For point cloud data with up to 800 points, the RIVET implementation completed minimal presentation computation in under 2 seconds, while Singular and Macaulay2 failed on larger instances.
• The cost of computing the semi-minimal presentation was often much higher than minimization, indicating that the main bottleneck lies in initial reduction.
• In homology degree 1, Singular's performance was dominated by the homology function call, with little difference between `res` and `fres` timings.
• The RIVET implementation outperformed both Singular and Macaulay2 across all tested data sets, including fractal and drag datasets, with speedups of over 100x on large instances.
• The algorithm enables interactive visualization in RIVET by efficiently computing the augmented arrangement and fibered barcodes via minimal presentations.