Generative Transfer for Entropic Optimal Transport with Unknown Costs
This paper addresses the practical challenge in Entropic Optimal Transport (EOT) where the underlying ground cost function is typically latent and unobserved. Rather than assuming a fixed geometric cost, we adopt a data-driven approach where a shared cost is revealed only through samples from a reference optimal coupling. The question is then: given samples from a reference optimal coupling, can we recover the optimal coupling for new marginals under the same latent cost? We introduce a generative transfer framework that recovers the optimal coupling for new marginals by utilizing an iterative path-wise tilting algorithm. Unlike static importance reweighting, this method evolves the coupling jointly with a marginal transport path, allowing mass to move beyond the reference support. We derive sample-level learning rules for these infinitesimal updates, which yield covariance-type evolution equations for the associated transport vector fields. By integrating this dynamics with Conditional Flow Matching (CFM), we produce a practical sampler for paired data. Finally, we provide theoretical guarantees establishing a global convergence rate of \mathcal{O}(\delta), ensuring the generated coupling converges to the target EOT plan in W_1 distance.