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 seamlessly 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）质心是一种在保持概率分布几何特性的前提下对其进行平均的数学严谨方法。简而言之，质心计算任务旨在基于给定的OT差异度量对一组概率分布进行平均。本文提出了一种新颖算法，用于逼近任意OT成本函数下的连续熵最优传输（EOT）质心。我们的方法建立在基于弱OT的EOT问题对偶重构基础上，该框架近期已受到机器学习社区的广泛关注。除了其新颖性之外，该方法还具有以下优势特性：（i）我们为所得解建立了质量边界；（ii）该方法与基于能量的模型（EBMs）学习过程无缝衔接，能够针对特定问题使用经过充分优化的算法；（iii）它提供了直观的优化方案，避免了极小极大化、强化学习及其他复杂技术手段。为验证方法有效性，我们考虑了若干低维场景和图像空间设置，包括非欧几里得成本函数。此外，我们研究了在预训练生成模型生成的图像流形上学习质心的实际任务，为现实世界应用开辟了新的研究方向。