Additive Partial Matchings Induced by Persistence Maps

TL;DR

The paper introduces the induced partial matching $\mathcal{P}_f$ as a new invariant for persistence maps $f: V \to U$, providing a richer characterization than the image module alone. It proves that $\mathcal{P}_f$ is additive over direct sums, enables recovery of the image module via a decomposition into $Z_{IJ}(f)$ components, and is computable in $O(M^3)$ time through a matrix-reduction scheme with respect to persistence bases. A key technical contribution is the chain-contraction framework that enables computing persistence maps between flag complexes even after independent edge collapses, dramatically speeding up computations. The authors implement the method (including edge-collapse acceleration) and demonstrate substantial runtime improvements on Vietoris–Rips filtrations, highlighting practical impact for topological data analysis of large or simplified complexes. The work also clarifies relationships with related invariants like $\chi_f$ and $\mathcal{M}_f$ and outlines future directions for extending to more general map configurations and continuous persistence settings.

Abstract

Persistent homology is a fundamental tool in Topological Data Analysis. The associated algebraic structure is the persistence module, a sequence of vector spaces connected by linear maps. Persistence modules admit a complete and fast-to-compute invariant known as the persistence diagram. However, this is no longer the case for maps between persistence modules (i.e. persistence maps). We propose a new invariant for persistence maps, consisting of a partial matching between the persistent diagrams of the domain and codomain modules. We show that this invariant is additive with respect to the direct sum decomposition of persistence maps, is more discriminative than the image invariant, and is computable in cubic time. Furthermore, we provide an implementation and demonstrate its efficiency by integrating it with edge collapse techniques for flag complexes (e.g., Vietoris-Rips complexes). As a key technical contribution, we describe how to induce a persistence map between two flag complexes that have been independently simplified via edge collapses, even when a direct simplicial map between them is no longer available.

Figures (7)

• Figure 1: Two examples of flag complexes with a dominated edge and the corresponding complex after the collapse. The dominated edges and their projection have been highlighted.
• Figure 2: Depiction of a pair of point samples $X \subset Z$ where the points from $X$ are plotted in red while points from $Z \setminus X$ are plotted in blue.
• Figure 3: Depiction of three cycles in $L'$ and in $K'$ corresponding to the matching $\mathcal{P}_{f}$. The flag complexes $L'$ and $K'$ are plotted in gray up to a fixed filtration value.
• Figure 4: Matrix associated to $\mathop{\mathrm{PH}}\nolimits_1(L')\rightarrow \mathop{\mathrm{PH}}\nolimits_1(K')$ (left) and corresponding induced matching (right). The order of columns (from left-to-right) and rows (from top-to-bottom) in $F$ corresponds to the order of plotted intervals on the right (from bottom-to-top). In addition, we mark (in dark blue) the intersection between matched intervals.
• Figure 5: Pair $X \subset Z$ of sampled points around a circle.

Theorems & Definitions (69)

• Example 1.1
• Example 2.1
• Theorem 2.2: ZC05CrawleyBoevey2015
• Example 2.3
• Definition 2.4
• Example 2.5
• Definition 2.6
• Lemma 2.7
• proof
• Lemma 2.8