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 {\delta} > 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 {\delta}, 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 {\delta}, 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.