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.

翻译：本文针对多参数持久性模块，引入了一种新的参数化拓扑描述符族，其形式为候选分解。我们从中识别出一个子族，称为近似分解，这些是可控的逼近，因为它们保持对角线条形码不变。接着，我们提出MMA（多持久性模块逼近）算法：该算法基于匹配函数，用于计算具有精度参数δ > 0的候选分解实例。通过设计，MMA能够处理任意数量的过滤，并具有有界的复杂度和运行时间。此外，我们证明了MMA的鲁棒性：当使用所谓的兼容匹配函数计算时，MMA能够产生近似分解（我们证明了对于n = 2个过滤，这样的匹配函数存在）。接下来，我们将重点限制在可分解为区间直和项的模块上。在这种情况下，兼容匹配函数始终存在，并且我们证明，对于足够小的δ，通过MMA使用此类兼容匹配函数获得的近似分解，其逼近误差（以标准交错距离和瓶颈距离衡量）以δ为界，并且当精度更小且为正时，误差可达到零。最后，我们通过实验证据验证了MMA在多个数据集上具有最先进的性能和运行时间。