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.

提出了一种新的算法，用于近似连续熵正交传输（EOT）重心，该方法建立在弱正交传输的基础上，具有优越性能，并与基于能量的模型（EBMs）学习流程无缝连接，为验证，考虑了几个低维场景和图像空间设置，包括非欧几里得成本函数，并探讨了在经过预训练的生成模型生成的图像流形上学习重心的实际任务，为实际应用开辟了新的方向。

能量引导的连续熵质心估计通用成本