Energy-Guided Continuous Entropic Barycenter Estimation for General Costs

TL;DR

This paper introduces a novel algorithm for approximating continuous Entropic OT barycenters with arbitrary costs, leveraging dual reformulation and energy-based models, applicable to low-dimensional and image data.

Contribution

It presents a new energy-guided method for continuous Entropic OT barycenter estimation that avoids complex min-max procedures and connects with energy-based models.

Findings

Provides quality bounds for solutions.

Effective in low-dimensional and image-space scenarios.

Enables learning barycenters on image manifolds.

Abstract

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. Our code is available at https://github.com/justkolesov/EnergyGuidedBarycenters.