Multi-parameter Module Approximation: an efficient and interpretable invariant for multi-parameter persistence modules with guarantees

TL;DR

This work addresses the challenge of extracting stable, interpretable descriptors for multi-parameter persistence modules. It introduces MMA, a scalable algorithm that computes δ-approximate decompositions as interval-summand candidates by matching one-dimensional slices along a δ-grid of diagonal lines, ensuring diagonal fibered barcodes are preserved within $2δ$. The authors prove approximation and stability guarantees for compatible matching functions, with stronger exact-recovery results in the interval-decomposable case when δ is sufficiently small. They further connect the method to vineyards updates for $n=2$ filtrations and demonstrate state-of-the-art performance on real and synthetic data, highlighting both interpretability and computational efficiency. The work lays groundwork for a potentially complete invariant formed by the collection of all compatible decompositions, and outlines future directions for extending guarantees to higher parameter counts and more general modules.

Abstract

In this article, we introduce a new parameterized family of topological descriptors, taking the form of candidate decompositions, for multi-parameter persistence modules, and we identify a subfamily of these descriptors, that we call approximate decompositions, that are controllable approximations, in the sense that they preserve diagonal barcodes. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm based on matching functions for computing instances of candidate decompositions with some precision parameter δ > 0. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Moreover, we prove the robustess of MMA: when computed with so-called compatible matching functions, we show that MMA produces approximate decompositions (and we prove that such matching functions exist for n = 2 filtrations). Next, we restrict the focus on modules that can be decomposed into interval summands. In that case, compatible matching functions always exist, and we show that, for small enough δ, the approximate decompositions obtained with such compatible matching functions by MMA have an approximation error (in terms of the standard interleaving and bottleneck distances) that is bounded by δ, and that reaches zero for an even smaller, positive precision. Finally, we present empirical evidence validating that MMA has state-of-the-art performance and running time on several data sets.

Figures (19)

• Figure 1: The different steps of MMA for computing a candidate decomposition of the module ${\mathbb{M} = k^{{ \hbox{\includesvg[width=2em]{symbols/solo_blue_rectangle2.svg}\xspace}}} \oplus k^{{ \hbox{\includesvg[width=2em]{symbols/solo_red_z.svg}\xspace}}}}$.
• Figure 2: Example of candidate decomposition computed by MMA on a point cloud filtered by both growing balls around the points (also called Čech filtration) and using the sublevel sets of codensity (or, equivalently, the superlevel sets of density), in homology dimension 1. One can see that there is a large lightgreen summand in the candidate decomposition on the right that corresponds to the cycle formed by the points amid outliers, which is also highlighted in lightgreen in the multi-parameter filtration on the left.
• Figure 3: Lower- and upper-boundaries of an interval in $\mathbb{R}^2$ (Definition \ref{['def:upper_lower_boundaries']}); and birthpoints and deathpoints $b_x^I$ and $d_x^I$ (Definition \ref{['def:birthpoints_deathpoints']}) of a point $x \in \mathbb{R}^2$.
• Figure 4: Two bars $[b_1,d_1)$ and $[b_2,d_2)$ of an interval module.
• Figure 5: Example of candidate decompositions for a $2$-interval module $\mathbb{I}$ with support in $\mathbb{R}^2$. (Left) Given the $L$-fibered barcode of $\mathbb{I}$, where $L$ is the family of the four black lines, we want to approximate $\mathbb{I}$ with an element of $S$, i.e., an interval module with the same fibered barcode. (Middle) When one further constrains the set $S$ by asking to have at most one corner between two consecutive endpoints of the fibered barcode, the whole set $S$ can be computed explicitly. (Right) The set $S$ can also be described as the set of intervals which have to go through the blue path, and which can arbitrarily choose between the red or green path at three different locations. Hence, the cardinality of $S$ is $2^3$.

Theorems & Definitions (73)

• Definition 1: Multi-parameter persistence module
• Definition 2: Pointwise finite dimensional module
• Definition 3
• Definition 4: Interleaving distance
• Definition 5: Bottleneck distance
• Definition 6: Interval
• Definition 7: Interval module, indicator module
• Definition 8: Discretely presented interval module
• Definition 9: Finitely presentable persistence module.
• Definition 10: Restrictions and slices