The theory of optimal transportation, dating back to Gaspard Monge’s work in 1781, continues to develop at pace as one of the fundamental mathematical theories with an ever- growing list of diverse applications in fields such as economics, computer vision, image processing and machine learning. A central challenge in many applications concerns finding a representative, or barycentric (probability) distribution, which provides some average description of a given set of distributions. The basic optimal transport approach to this problem is to find the barycenter by minimizing the sum of weighted two-marginal optimal transport costs between the barycenter and each input distributions. In a seminal contribution by Agueh and Carlier [1], it was subsequently shown that an equivalent and computationally favourable approach is to instead solve a single least-cost multi-marginal optimal transport problem.

If the input distributions do not all have equal mass, an unbalanced barycenter can be found via a recourse to the emerging theory of unbalanced optimal transportation. This, however, can be done in a number of ways, depending on how one penalises mass deviations, what cost function is employed and whether one wishes to consider the conic formulation — see the detailed discussion in [2].

In this talk, I will introduce the above ideas in an accessible manner, followed by presenting several results on how to recover the celebrated least-cost multi-marginal formulation of Agueh and Carlier in the unbalanced setting [3].

[1] M. Agueh and G. Carlier. Barycenters in the Wasserstein space. SIAM Journal on Mathematical Analysis 43.2 (2011), pp. 904–924.

[2] M. Liero, A. Mielke, and G. Savaré. Optimal entropy-transport problems and a new Hellinger– Kantorovich distance between positive measures. Inventiones mathematicae 211.3 (2018), pp. 969– 1117.

[3] M. Buze. Constrained Hellinger-Kantorovich barycenters: least-cost soft and conic multi-marginal formulations. arXiv e-prints, 2402.11268, 2024 (to appear in SIAM Journal on Mathematical Analysis).