The interleaving distance is arguably the most widely used metric in topological data analysis (TDA) due to its applicability to a wide array of inputs of interest, such as (multiparameter) persistence modules, Reeb graphs, merge trees, and zigzag modules. However, computation of the interleaving distance in the vast majority of this settings is known to be NP-hard, limiting its use in practical settings. Inspired by the work of Chambers et al. on the interleaving distance for mapper graphs, we solve a more general problem bounding the interleaving distance between generalized persistence modules on concrete categories via a loss function. This loss function measures how far an assignment, which can be thought of as an interleaving that might not commute, is from defining a true interleaving. We give settings for which the loss can be computed in polynomial time, including for certain assumptions on $k$-parameter persistence modules.

翻译：交错距离无疑是拓扑数据分析（TDA）中应用最广泛的度量，因其适用于多种重要输入，例如（多参数）持久性模、Reeb图、合并树以及之字形模。然而，在绝大多数此类设定中，交错距离的计算已知是NP难的，这限制了其在实际场景中的应用。受Chambers等人关于mapper图交错距离研究的启发，我们通过引入一个损失函数，解决了一个更普遍的问题：在具体范畴上界定广义持久性模之间的交错距离。该损失函数度量了一个赋值（可视为可能不满足交换性的交错）距离定义一个真正交错有多远。我们给出了损失可在多项式时间内计算的若干设定，包括对$k$参数持久性模的某些假设情形。