Parallel computation of interval bases for persistence module decomposition

TL;DR

This work addresses the computation of interval bases for persistence module decompositions by introducing a parallel, kernel-flag based algorithm that avoids constructing a presentation and its Smith normal form. The method processes each persistence step independently, builds an interval basis $\mathscr{V}$, and yields the corresponding persistence diagram without explicit matrix reductions. It provides a detailed complexity analysis with an output-aware cost that can outperform SNF in typical barcodes, and it specializes to persistent homology modules, including constructions via harmonics using the Hodge Laplacian. The approach extends naturally to simplicial complexes and filtered complexes, and it enables parallel tracking of harmonic representatives, suggesting practical benefits for large-scale TDA and harmonic analysis on filtered data.

Abstract

A persistence module $M$, with coefficients in a field $\mathbb{F}$, is a finite-dimensional linear representation of an equioriented quiver of type $A_n$ or, equivalently, a graded module over the ring of polynomials $\mathbb{F}[x]$. It is well-known that $M$ can be written as the direct sum of indecomposable representations or as the direct sum of cyclic submodules generated by homogeneous elements. An interval basis for $M$ is a set of homogeneous elements of $M$ such that the sum of the cyclic submodules of $M$ generated by them is direct and equal to $M$. We introduce a novel algorithm to compute an interval basis for $M$. Based on a flag of kernels of the structure maps, our algorithm is suitable for parallel or distributed computation and does not rely on a presentation of $M$. This algorithm outperforms the approach via the presentation matrix and Smith Normal Form. We specialize our parallel approach to persistent homology modules, and we close by applying the proposed algorithm to tracking harmonics via Hodge decomposition.

Figures (8)

• Figure 1: A 3-step filtration by sublevel sets, for the $z$ coordinate, of a tilted and triangulated half torus.
• Figure 2: In red, two representative $1$-cycles. Their homology classes form a minimal system of generators of the persistent homology module of the filtration in \ref{['fig:example_filtration']}. These representatives do not induce an interval basis.
• Figure 3: In red, a different choice of representative $1$-cycles. Their homology classes are a different choice of generators for the same persistence module as in \ref{['fig:example_classical']}. However, these generators do induce an interval basis.
• Figure 4: Harmonic representatives via the interval basis algorithms
• Figure 5: An example of a vertex collapse inducing a chain map $f=(f_k)$, where step 2 is obtained from step 1 by identifying vertices $2$ and $4$. Coefficients of $1$-chains of possible degree-1 homology representatives are depicted with the same color. The figure shows, at step 1 in red $z_1 = [1,2] + [2,3] - [1,3]$, in green $z_2 = [1,3] + [3,4] - [1,4]$, in blue $z = [1,2] + [2,3] + [3,4] - [1,4]$; at step 2 in red $\omega_1 = [1,2] + [2,3] - [1,3]$, in green $\omega_2 = [1,3] - [2,3] - [1,2]$. The $1$-component $f_1$ of the chain map $f$ sends $z_1$ to $\omega_1$, and $z_2$ to $\omega_2$. Hence, the image of $z=z_1+z_2$ is trivial.

Theorems & Definitions (46)

• Definition 1.1
• Remark 2.1
• Definition 2.2
• Proposition 2.3
• proof
• Lemma 4.1
• proof
• Definition 4.2
• Lemma 4.3
• proof