Variational Entropic Optimal Transport
This addresses a computational bottleneck for researchers and practitioners using EOT for domain translation problems, offering a more efficient optimization method without requiring MCMC simulations.
The paper tackles the computational inefficiency of optimizing the weak dual entropic optimal transport (EOT) objective by proposing Variational Entropic Optimal Transport (VarEOT), which reformulates the intractable log-partition term into a tractable minimization, resulting in competitive or improved translation quality in experiments on synthetic data and unpaired image-to-image translation.
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.