# On optimal transport of matrix-valued measures

arXiv: https://arxiv.org/abs/1808.05064 · HTML: https://mlsanity.com/paper/1808.05064/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 15 Aug 2018.

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

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- [Two-Scale Gradient Descent Ascent Dynamics Finds Mixed Nash Equilibria of Continuous Games: A Mean-Field Perspective](https://mlsanity.com/paper/2212.08791/) (2022)
- [Fisher-Rao Gradient Flow: Geodesic Convexity and Functional Inequalities](https://mlsanity.com/paper/2407.15693/) (2024)
- [A deep learning framework for geodesics under spherical Wasserstein-Fisher-Rao metric and its application for weighted sample generation](https://mlsanity.com/paper/2208.12145/) (2022)

## Abstract (arXiv)

We suggest a new way of defining optimal transport of positive-semidefinite matrix-valued measures. It is inspired by a recent rendering of the incompressible Euler equations and related conservative systems as concave maximization problems. The main object of our attention is the Kantorovich-Bures metric space, which is a matricial analogue of the Wasserstein and Hellinger-Kantorovich metric spaces. We establish some topological, metric and geometric properties of this space, which includes the existence of the optimal transportation path.