🤖 AI Summary
This work addresses the multi-marginal optimal transport (MOT) problem under graph-structured cost functions—arising in traffic flow control, Wasserstein barycenter/regression, and hidden Markov inference. We propose the first parallelizable MOT framework for tree-structured graphs and extend it to general graphs via an enhanced junction tree decomposition, enabling parallel block-coordinate updates of dual variables. Theoretical analysis shows substantially reduced computational complexity; numerical experiments demonstrate superior speedup and state-of-the-art computational efficiency. Our core contributions are: (1) establishing a systematic connection between graph-structured MOT and probabilistic graphical models; (2) designing the first strictly parallelizable MOT algorithm for tree-structured costs; and (3) introducing junction tree methods to general-graph MOT optimization—opening a new pathway toward scalable, efficient MOT solvers.
📝 Abstract
We study multi-marginal optimal transport (MOT) problems where the underlying cost has a graphical structure. These graphical multi-marginal optimal transport problems have found applications in several domains including traffic flow control and regression problems in the Wasserstein space. MOT problem can be approached through two aspects: a single big MOT problem, or coupled minor OT problems. In this paper, we focus on the latter approach and demonstrate it has efficiency gain from the parallelization. For tree-structured MOT problems, we introduce a novel parallelizable algorithm that significantly reduces computational complexity. Additionally, we adapt this algorithm for general graphs, employing the modified junction trees to enable parallel updates. Our contributions, validated through numerical experiments, offer new avenues for MOT applications and establish benchmarks in computational efficiency.