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.

翻译：具有二次成本的连续空间中的熵最优传输（EOT）是解决域转换问题的经典工具。在实践中，最近的方法优化了一个依赖于单一势函数的弱对偶EOT目标，但由于难以处理的log-配分项，这样做在计算上并不高效。现有方法通常通过以下两种方式之一解决这一障碍：要么通过显著限制传输族以获得闭式归一化（通过高斯混合参数化），要么使用需要基于模拟训练过程的通用神经参数化。我们提出了变分熵最优传输（VarEOT），它基于log-配分项 $\log \mathbb{E}[\exp(\cdot)]$ 的一个精确变分重构，将其表示为一个在辅助正归一化器上的可处理的最小化问题。这产生了一个可通过随机梯度优化的可微学习目标，并避免了训练期间进行MCMC模拟的必要性。我们提供了理论保证，包括有限样本泛化界和通用函数逼近下的近似结果。在合成数据和未配对图像到图像转换上的实验证明了具有竞争力或改进的转换质量，而在使用相同弱对偶EOT目标的求解器内部的比较支持了所提出的优化原理的益处。