Barycenters for the Hellinger--Kantorovich distance over ${\mathbb{R}}^d$

Abstract

We study the barycenter of the Hellinger--Kantorovich metric over non-negative measures on compact, convex subsets of ${\mathbb{R}}^d$. The article establishes existence, uniqueness (under suitable assumptions) and equivalence between a coupled-two-marginal and a multi-marginal formulation. We analyze the HK barycenter between Dirac measures in detail, and find that it differs substantially from the Wasserstein barycenter by exhibiting a local `clustering' behaviour, depending on the length scale of the input measures. In applications it makes sense to simultaneously consider all choices of this scale, leading to a 1-parameter family of barycenters. We demonstrate the usefulness of this family by analyzing point clouds sampled from a mixture of Gaussians and inferring the number and location of the underlying Gaussians.

arXiv commentMinor changes to keep consistent with journal version

References in corpus (4)

• Optimal Entropy-Transport problems and a new Hellinger-Kantorovich distance between positive measures
• Optimal-transport formulation of electronic density-functional theory
• Optimal transport in competition with reaction: the Hellinger-Kantorovich distance and geodesic curves
• Smoothing of transport plans with fixed marginals and rigorous semiclassical limit of the Hohenberg-Kohn functional

Cited by in corpus (2)

• Unbalanced Multi-Marginal Optimal Transport
• Constrained Hellinger-Kantorovich barycenters: least-cost soft and conic multi-marginal formulations