A geometric graph is an abstract graph along with an embedding of the graph into the Euclidean plane which can be used to model a wide range of data sets. The ability to compare and cluster such objects is required in a data analysis pipeline, leading to a need for distances or metrics on these objects. In this work, we study the interleaving distance on geometric graphs, where functor representations of data can be compared by finding pairs of natural transformations between them. However, in many cases, particularly those of the set-valued functor variety, computation of the interleaving distance is NP-hard. For this reason, we take inspiration from the work of Robinson to find quality measures for families of maps that do not rise to the level of a natural transformation. Specifically, we call collections $\phi = \{\phi_U\mid U\}$ and $\psi = \{\psi_U\mid U\}$ which do not necessarily form a true interleaving an \textit{assignment}. In the case of embedded graphs, we impose a grid structure on the plane, treat this as a poset endowed with the Alexandroff topology $K$, and encode the embedded graph data as functors $F: \mathbf{Open}(K) \to \mathbf{Set}$ where $F(U)$ is the set of connected components of the graph inside of the geometric realization of the set $U$. We then endow the image with the extra structure of a metric space and define a loss function $L(\phi,\psi)$ which measures how far the required diagrams of an interleaving are from commuting. Then for a pair of assignments, we use this loss function to bound the interleaving distance, with an eye toward computation and approximation of the distance. We expect these ideas are not only useful in our particular use case of embedded graphs, but can be extended to a larger class of interleaving distance problems where computational complexity creates a barrier to use in practice.

翻译：几何图是一类抽象图及其到欧几里得平面的嵌入，可用于建模多种数据集。在数据分析流程中，需要对此类对象进行比较和聚类，因此需要定义对象间的距离或度量。本文研究几何图上的交错距离，其中可通过寻找数据函子表示之间的自然变换对进行数据比较。然而，在诸多情形中（特别是集值函子类），交错距离的计算属于NP困难问题。为此，我们借鉴Robinson的研究思路，为未达到自然变换级别的映射族建立质量度量。具体而言，我们将未必构成真正交错的集合族$\phi = \{\phi_U\mid U\}$和$\psi = \{\psi_U\mid U\}$称为**指派**。对于嵌入图的情形，我们在平面上建立网格结构，将其视为赋予Alexandroff拓扑$K$的偏序集，并将嵌入图数据编码为函子$F: \mathbf{Open}(K) \to \mathbf{Set}$，其中$F(U)$是图在集合$U$几何实现内部的连通分支集。进而赋予该像集以度量空间结构，并定义损失函数$L(\phi,\psi)$用以度量交错所需图表偏离交换的程度。基于指派对，我们利用该损失函数界定交错距离，旨在于距离的计算与近似应用。预期这些思想不仅适用于嵌入图的特例，还可扩展至因计算复杂性阻碍实际应用的一类更广泛的交错距离问题。