In this article, we introduce a new parameterized family of topological invariants, taking the form of candidate decompositions, for multi-parameter persistence modules. We prove that our candidate decompositions are controllable approximations: when restricting to modules that can be decomposed into interval summands, we establish theoretical results about the approximation error between our candidate decompositions and the true underlying module in terms of the standard interleaving and bottleneck distances. Moreover, even when the underlying module does not admit such a decomposition, our candidate decompositions are nonetheless stable invariants; small perturbations in the underlying module lead to small perturbations in the candidate decomposition. Then, we introduce MMA (Multipersistence Module Approximation): an algorithm for computing stable instances of such invariants, which is based on fibered barcodes and exact matchings, two constructions that stem from the theory of single-parameter persistence. By design, MMA can handle an arbitrary number of filtrations, and has bounded complexity and running time. Finally, we present empirical evidence validating the generalization capabilities and running time speed-ups of MMA on several data sets.

翻译：本文提出了一族新的参数化拓扑不变量，其形式为候选分解，适用于多参数持久性模。我们证明这些候选分解是可控的近似：当限制在可分解为区间直和的模时，我们建立了理论结果，表明在标准交错距离和瓶颈距离下，候选分解与真实底层模之间的近似误差具有可控性。此外，即使底层模不承认此类分解，候选分解仍具有稳定性；底层模的微小扰动仅导致候选分解的微小变化。随后，我们引入MMA（多持续模近似）算法：一种计算此类不变量稳定实例的算法，该算法基于纤维化条形码与精确匹配——这两个构造源自单参数持久性理论。通过设计，MMA可处理任意数量的过滤，并具有有界复杂度与运行时间。最后，我们通过多个数据集上的实证证据，验证了MMA的泛化能力与运行时间加速效果。