  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.
