# Entropy-Transport distances between unbalanced metric measure spaces

arXiv: https://arxiv.org/abs/2009.10636 · HTML: https://mlsanity.com/paper/2009.10636/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 22 Sep 2020.

## Built on by (most cited first, 2 of 2)

- [Weisfeiler-Lehman meets Gromov-Wasserstein](https://mlsanity.com/paper/2202.02495/) (2022)
- [Unbalanced CO-Optimal Transport](https://mlsanity.com/paper/2205.14923/) (2022)

## Abstract (arXiv)

Inspired by the recent theory of Entropy-Transport problems and by the D-distance of Sturm on normalised metric measure spaces, we define a new class of complete and separable distances between metric measure spaces of possibly different total mass. We provide several explicit examples of such distances, where a prominent role is played by a geodesic metric based on the Hellinger-Kantorovich distance. Moreover, we discuss some limiting cases of the theory, recovering the "pure transport" D-distance and introducing a new class of "pure entropic" distances. We also study in detail the topology induced by such Entropy-Transport metrics, showing some compactness and stability results for metric measure spaces satisfying Ricci curvature lower bounds in a synthetic sense.