Universal Mappings and Analysis of Functional Data on Geometric Domains

One can model a data set as a metric space or a metric measure space with a function, which I am going to call a field. For example, a weighted network with labeled vertices can be modeled as a metric measure space with a function by endowing the set of vertices with the shortest path distance and normalized counting measure. As data sets are noisy, constructions applied to them should be stable, in the sense that similar data sets should produce similar outputs. This requires a method to measure the degree of similarity between the objects used for modeling data sets, in particular fields.

In this talk, I am going to introduce analogues of Gromov-Hausdorff and Gromov-Wasserstein distances for fields, and state some of their properties. Then, I will obtain a geometric representation of isomorphism classes of fields under these metrics through the construction of Urysohnn universal fields.

(Joint work with S. Anbouhi and W. Mio.)