Data consisting of a graph with a function mapping into $\R^d$ arise in many data applications, encompassing structures such as Reeb graphs, geometric graphs, and knot embeddings. As such, the ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances between them. In this work, we study the interleaving distance on discretization of these objects, $\R^d$-mapper graphs, where functor representations of the data can be compared by finding pairs of natural transformations between them. However, in many cases, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from recent work by Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation, called assignments. We then endow the functor images with the extra structure of a metric space and define a loss function which measures how far an assignment is from making the required diagrams of an interleaving commute. Finally we show that the computation of the loss function is polynomial with a given assignment. We believe this idea is both powerful and translatable, with the potential to provide approximations and bounds on interleavings in a broad array of contexts.

翻译：由具有映射到$\R^d$的函数构成的图数据出现在许多数据应用中，涵盖Reeb图、几何图和纽结嵌入等结构。因此，在数据分析流程中需要比较和聚类此类对象，从而产生了对这些对象之间距离的需求。本文研究这些对象的离散化形式——$\R^d$-Mapper图上的交错距离，其中可以通过寻找函子表示之间的自然变换对来比较数据。然而，在许多情况下，交错距离的计算是NP难的。为此，我们从Robinson近期的工作中汲取灵感，为那些未达到自然变换层次的映射族（称为赋值）寻找质量度量。随后，我们为函子像赋予度量空间的额外结构，并定义一个损失函数，该函数衡量赋值在多大程度上使得交错所需的示意图可交换。最后证明在给定赋值下损失函数的计算是多项式时间的。我们相信这一想法既强大又可推广，具有在广泛情境中提供交错近似与边界的潜力。