On a Generalization of Wasserstein Distance and the Beckmann Problem to Connection Graphs

We propose a model of optimal parallel transport between vector fields on a connection graph, which consists of a weighted graph along with a map from its edges to an orthogonal group. Inspired by the well-known equivalence of 1-Wasserstein distance and minimum cost flows on standard graphs, we consider two versions of this problem: a minimum norm vector-valued flow problem with divergence constraints reflective of the connection structure of the graph; and a modified version which incorporates both quadratic regularization and a relaxation of the divergence constraint. Our theoretical contributions include: conditions for feasibility and computation of the Lagrangian dual problem for both problems, and duality correspondence for the relaxed-regularized version. Example applications of the model including transport between color images, vector field interpolation, and unsupervised clustering of vector field-valued data (in this case hurricane trajectory data) are also considered.