🤖 AI Summary
This work addresses optimal parallel transport of vector fields on connection graphs, introducing the first vector-valued optimal transport (OT) framework for such geometric structures. Methodologically, it formulates the problem as a minimum-norm vector flow subject to a connection-divergence constraint; it further proposes quadratic regularization and constraint relaxation variants, thereby generalizing both the Wasserstein distance and the Beckmann problem to weighted graphs endowed with affine connections. Theoretically, it establishes a rigorous Lagrangian duality framework; algorithmically, it integrates graph-theoretic methods, orthogonal-group-valued connection maps, and vector-valued flow optimization. Experiments on color image transfer, vector field interpolation, and unsupervised clustering of hurricane trajectories demonstrate substantial improvements in geometric awareness. Key contributions include: (i) the first theoretical OT framework for vector fields on connection graphs; (ii) a computationally tractable regularized model; and (iii) empirical validation of its efficacy for non-Euclidean vector data analysis.
📝 Abstract
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.
