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)重心提供了一种数学上严谨的概率分布平均化方法，同时能够捕捉其几何特性。简言之，重心任务是基于给定OT差异度，对一组概率分布进行平均化。我们提出了一种新颖算法，用于近似任意OT代价函数下的连续熵正则化OT(EOT)重心。该方法基于弱OT理论对EOT问题的对偶重构，该理论近期引起了机器学习界的关注。除创新性外，我们的方法具备多项优越性质：(i)建立了恢复解的质量边界；(ii)该方法与基于能量的模型(EBMs)学习流程无缝衔接，能利用经过充分优化的算法解决目标问题；(iii)提供了直观的优化方案，避免了最小-最大、强化学习及其他复杂技术技巧。为验证有效性，我们考虑了若干低维场景和图像空间设置，包括非欧几里得代价函数。此外，我们还探索了在预训练生成模型构建的图像流形上学习重心的实际任务，为真实世界应用开辟了新方向。