Computing Barycentres of Measures for Generic Transport Costs
This work extends the applicability of barycentre computation to a broader range of problems, but it is incremental as it builds on existing fixed-point methods.
The paper tackles the challenge of computing Wasserstein barycentres for generic transport costs and probability measures by generalizing a fixed-point method, demonstrating convergence and illustrating its numerical behavior on various problems.
Wasserstein barycentres represent average distributions between multiple probability measures for the Wasserstein distance. The numerical computation of Wasserstein barycentres is notoriously challenging. A common approach is to use Sinkhorn iterations, where an entropic regularisation term is introduced to make the problem more manageable. Another approach involves using fixed-point methods, akin to those employed for computing Fréchet means on manifolds. The convergence of such methods for 2-Wasserstein barycentres, specifically with a quadratic cost function and absolutely continuous measures, was studied by Alvarez-Esteban et al. (2016). In this paper, we delve into the main ideas behind this fixed-point method and explore how it can be generalised to accommodate more diverse transport costs and generic probability measures, thereby extending its applicability to a broader range of problems. We show convergence results for this approach and illustrate its numerical behaviour on several barycentre problems.