Variational Entropic Optimal Transport

Entropic optimal transport (EOT) in continuous spaces with quadratic cost is a classical tool for solving the domain translation problem. In practice, recent approaches optimize a weak dual EOT objective depending on a single potential, but doing so is computationally not efficient due to the intractable log-partition term. Existing methods typically resolve this obstacle in one of two ways: by significantly restricting the transport family to obtain closed-form normalization (via Gaussian-mixture parameterizations), or by using general neural parameterizations that require simulation-based training procedures. We propose Variational Entropic Optimal Transport (VarEOT), based on an exact variational reformulation of the log-partition $\log \mathbb{E}[\exp(\cdot)]$ as a tractable minimization over an auxiliary positive normalizer. This yields a differentiable learning objective optimized with stochastic gradients and avoids the necessity of MCMC simulations during the training. We provide theoretical guarantees, including finite-sample generalization bounds and approximation results under universal function approximation. Experiments on synthetic data and unpaired image-to-image translation demonstrate competitive or improved translation quality, while comparisons within the solvers that use the same weak dual EOT objective support the benefit of the proposed optimization principle.

Many AI tasks involve turning one kind of data into another — for example, transforming a photo of a man into a plausible photo of a woman, without ever seeing matched pairs. The mathematical framework for moving one collection of data onto another as "cheaply" as possible is called optimal transport; its entropic version adds controlled randomness, so a single input can map to many plausible outputs. Existing methods face an uncomfortable tradeoff: flexible ones must run a slow, finicky random-sampling routine at every training step, while faster ones stay fast only by forcing the mapping into a rigid, predefined shape. The technical culprit is a normalizing quantity that is generally impossible to compute exactly. We introduce VarEOT, which sidesteps this obstacle through an exact reformulation: the awkward quantity is replaced by a helper function learned jointly with the main model using ordinary optimization. Training therefore becomes fast, requires no sampling, and places no artificial restriction on the mapping. We also prove the method recovers the true solution given enough data and expressive models, and confirm this on synthetic data, biological single-cell measurements, and face-image translation — matching or surpassing prior methods while running several times faster.