Energy-Guided Continuous Entropic Barycenter Estimation for General Costs

2023-10-02

arXiv.org

AI Summary

Computing geometrically faithful barycenters of probability distributions in non-Euclidean spaces remains challenging due to the intrinsic complexity of distributional geometry under arbitrary cost functions. Method: This paper proposes a novel continuous entropy-regularized optimal transport (EOT) barycenter estimation framework applicable to arbitrary ground costs. Leveraging weak OT duality theory, it establishes an energy-guided optimization paradigm that enables end-to-end coupling of EOT barycenter estimation with energy-based model (EBM) training. Contribution/Results: Theoretically, it derives quality bounds on the estimated barycenter and eliminates unstable mechanisms such as min-max adversarial training and REINFORCE-based gradient estimation. Practically, it delivers a gradient-driven, geometry-aware, and numerically robust optimization procedure. Experiments across low-dimensional distributions, image spaces (including non-Euclidean costs), and pretrained generative manifolds demonstrate substantial improvements in both geometric fidelity and computational stability of barycenter estimation.

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.

Problem

• Optimal Transport Barycenter
• Complex Cost Function
• Non-Euclidean Space

Innovation

• Optimal Transport Barycenter
• Weak OT Dual Perspective
• Non-Euclidean Cost Functions