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This paper introduces Potential Matching Optimal Transport (PMOT), a novel framework for $p$-cost optimal transport that utilizes continuous normalizing flows (CNFs) to parameterize the velocity field through a scalar potential. The method employs a self-induced matching loss to ensure exact terminal distribution matching while maintaining flexibility in the model's endpoints. The key finding is that PMOT achieves zero-loss exactness, allowing for the recovery of the $p$-optimal transport map and dynamics, validated through synthetic benchmarks and competitive performance on high-dimensional data tasks.
Zero-loss exactness in optimal transport dynamics could revolutionize how we approach matching problems in high-dimensional spaces.
We introduce Potential Matching Optimal Transport (PMOT), a potential-flow framework for general $p$-cost optimal transport with $c_p(x,y)=\|x-y\|^p$. PMOT parameterizes the CNF velocity field with a scalar potential in the generalized Benamou--Brenier form for the chosen exponent $p$. It trains the potential gradient with a self-induced matching loss along straight bridges determined by the model's own endpoints, while allowing flexible terminal distribution matching. Our main result establishes zero-loss exactness: under the stated regularity, exact terminal matching, and uniqueness assumptions, any zero-loss solution satisfies the generalized Benamou--Brenier optimality system and recovers the corresponding $p$-optimal transport map and dynamics. On synthetic benchmarks, PMOT learns $p$-specific maps that agree with the corresponding $p$-matched OT references. It also remains competitive as a likelihood-based density model on high-dimensional tabular data, and an MMD-based color transformation experiment demonstrates flexible sample-based terminal matching.
