In this paper, we explore the untapped intersection of the graph connection Laplacian and discrete optimal transport to propose a novel framework for studying optimal parallel transport between vector fields on graphs. Our study establishes feasibility conditions for the resulting convex optimization problem on connection graphs. Furthermore, we establish strong duality for the so-called connection Beckmann problem, and extend our analysis to encompass strong duality and duality correspondence for a quadratically regularized variant. Then, we implement the model across a selection of several examples leveraging both synthetic and real-world datasets drawn from computer graphics and meteorology.

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