Variational Entropic Optimal Transport

TL;DR

This paper introduces VarEOT, a novel variational reformulation of entropic optimal transport that enables efficient, gradient-based training without simulation, improving domain translation tasks.

Contribution

It proposes an exact variational reformulation of the log-partition term in EOT, allowing efficient optimization with stochastic gradients and theoretical guarantees.

Findings

Competitive translation quality on synthetic and image data

Avoids MCMC simulations during training

Supports theoretical generalization bounds

Abstract

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.