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

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.