  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.
