Variational Entropic Optimal Transport

2026-02-02

AI Summary

This work addresses the computational inefficiency in entropy-regularized optimal transport (EOT) over continuous spaces, which arises from the intractability of the log-partition function. The authors propose a variational reformulation that yields, for the first time, an exact variational representation of this partition function, recasting it as a differentiable minimization problem over auxiliary normalized variables. This approach eliminates the need for conventional assumptions such as Gaussian mixture approximations or reliance on MCMC sampling, thereby enabling end-to-end training. By integrating neural function approximation with stochastic gradient optimization, the method yields an efficient and scalable differentiable EOT solver. Experiments on both synthetic data and unpaired image-to-image translation tasks demonstrate that the proposed framework matches or exceeds the performance of existing methods, confirming its effectiveness.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Reasoning under Uncertainty: Stochastic Optimization
• Computer Vision: Learning & Optimization for CV

Application Category

• Graph Algorithms and Modeling for the Web: Graph embeddings and representation learning for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Economics, Online Markets and Human Computation: Data quality aspects of human-annotated datasets

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.

Problem

• Entropic Optimal Transport
• Domain Translation
• Log-partition Function
• Computational Efficiency
• Continuous Spaces

Innovation

• Variational Entropic Optimal Transport
• log-partition variational reformulation
• differentiable optimization
• stochastic gradient training
• unpaired image-to-image translation