# Barycenters for the Hellinger--Kantorovich distance over ℝd

arXiv: https://arxiv.org/abs/1910.14572 · HTML: https://mlsanity.com/paper/1910.14572/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 Oct 2019.

## Built on by (most cited first, 4 of 4)

- [Robust Barycenter Estimation using Semi-Unbalanced Neural Optimal Transport](https://mlsanity.com/paper/2410.03974/) (2024)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Conic Formulations of Transport Metrics for Unbalanced Measure Networks and Hypernetworks](https://mlsanity.com/paper/2508.10888/) (2025)
- [Lower Bounds on Adversarial Robustness for Multiclass Classification with General Loss Functions](https://mlsanity.com/paper/2510.01969/) (2025)

## Abstract (arXiv)

We study the barycenter of the Hellinger--Kantorovich metric over non-negative measures on compact, convex subsets of ℝd. 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.