Optimal transport (OT) barycenters are a mathematically grounded way of averaging probability distributions while capturing their geometric properties. In short, the barycenter task is to take the average of a collection of probability distributions w.r.t. given OT discrepancies. We propose a novel algorithm for approximating the continuous Entropic OT (EOT) barycenter for arbitrary OT cost functions. Our approach is built upon the dual reformulation of the EOT problem based on weak OT, which has recently gained the attention of the ML community. Beyond its novelty, our method enjoys several advantageous properties: (i) we establish quality bounds for the recovered solution; (ii) this approach seemlessly interconnects with the Energy-Based Models (EBMs) learning procedure enabling the use of well-tuned algorithms for the problem of interest; (iii) it provides an intuitive optimization scheme avoiding min-max, reinforce and other intricate technical tricks. For validation, we consider several low-dimensional scenarios and image-space setups, including non-Euclidean cost functions. Furthermore, we investigate the practical task of learning the barycenter on an image manifold generated by a pretrained generative model, opening up new directions for real-world applications.

翻译：最优传输（OT）重心是一种在数学上严谨的概率分布平均化方法，能够同时捕捉其几何特性。简言之，重心任务是根据给定的OT差异度对一组概率分布进行平均。针对任意OT代价函数，我们提出了一种新型连续熵OT（EOT）重心近似算法。该方法基于弱OT理论中EOT问题的对偶重构——这一理论近期引起了机器学习界的广泛关注。我们的方法除创新性外，还具备若干优越特性：（i）建立了恢复解的质量界限；（ii）该方法可与基于能量的模型（EBMs）学习流程无缝衔接，从而能够针对目标问题运用经过充分调优的算法；（iii）它提供了直观的优化方案，避免了min-max、REINFORCE及其他复杂技术技巧。在验证环节，我们考虑了多个低维场景和图像空间设置，包括非欧几里得代价函数。此外，我们还探究了在预训练生成模型生成的图像流形上学习重心的实际任务，这为现实应用开辟了新方向。