Constrained Hellinger–Kantorovich Barycenters : Least-Cost Soft and Conic Multimarginal Formulations

Buze, Maciej (2025) Constrained Hellinger–Kantorovich Barycenters : Least-Cost Soft and Conic Multimarginal Formulations. SIAM Journal on Mathematical Analysis, 57 (1). pp. 495-519.

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Abstract

We show that the problem of finding the barycenter in the Hellinger–Kantorovich setting admits a least-cost soft multimarginal formulation, provided that a one-sided hard marginal constraint is introduced. The constrained approach is then shown to admit a conic multimarginal reformulation based on defining a single joint multimarginal perspective cost function in the conic multimarginal formulation, as opposed to separate two-marginal perspective cost functions for each two-marginal problem in the coupled-two-marginal formulation, as was studied previously in the literature. We further establish that, as in the Wasserstein metric, the recently introduced framework of unbalanced multimarginal optimal transport can be reformulated using the notion of the least cost. Subsequently, we discuss an example when input measures are Dirac masses and numerically solve an example for Gaussian measures. Finally, we also explore why the constrained approach can be seen as a natural extension of a Wasserstein space barycenter to the unbalanced setting.