Representing two-parameter persistence modules via graphcodes

TL;DR

The paper develops compressed graphcodes as a compact, complete descriptor for two-parameter persistence modules, establishing equivalence with minimal presentations and revealing structural connections: connected components correspond to module summands and directed-path components correspond to intervals. It provides a practical graphcode pipeline, including a cubic-time algorithm to convert presentations to graphcodes and a simple expansion mechanism to recover presentations, with compression reducing graph size to $O(n^2)$. A key theoretical advance is a graphcode-based criterion and $O(N^4)$ time interval-decomposability test for distinctly-graded presentations, improving previous bounds and enabling faster preliminary decompositions. Empirically, compression yields substantial performance gains in some cases and modest or mixed gains in others, while the graphcode view serves as a useful preprocessing step for downstream tasks like indecomposable decomposition via aida. The work opens avenues for graphcode-driven algorithms directly operating on the combinatorial representation and for extending the framework to broader posets.

Abstract

Graphcodes were recently introduced as a technique to employ two-parameter persistence modules in machine learning tasks (Kerber and Russold, NeurIPS 2024). We show in this work that a compressed version of graphcodes yields a description of a two-parameter module that is equivalent to a presentation of the module. This alternative representation as a graph allows for a simple translation between combinatorics and algebra: connected components of the graphcode correspond to summands of the module and isolated paths correspond to intervals. We demonstrate that graphcodes are useful in practice by speeding-up the task of decomposing a module into indecomposable summands. Also, the graphcode viewpoint allows to devise a simple algorithm to decide whether a persistence module is interval-decomposable in $O(n^4)$ time, which improves on the previous bound of $O(n^{2ω+1})$.

Figures (5)

• Figure 1: Construction of graphcode from matrix representations.
• Figure 2: Graphcode of interval module supported on the red shaded area.
• Figure 3: Graphcode (left), compressed graphcode (right).
• Figure 4: A presentation matrix (right) constructed from a graphcode (left).
• Figure 5: A graphcode that is a disjoint union of directed paths up to index $3$. At index $3$ we can not add column $1$ to column $2$ since $[2,6)\centernot{\lhd}[3,5)$ at index $2$. But we can add column $1$ to column $3$ since $[1,4)\cap[5,7)=\emptyset$.

Theorems & Definitions (14)

• Definition 1: Graphcode
• Proposition 2
• Proposition 3
• Proposition 4
• Proposition 5
• Corollary 6
• Proposition 7
• Definition 8: generalized Graphcode
• Theorem 9
• Theorem 10