# Matrix-valued Monge-Kantorovich Optimal Mass Transport

arXiv: https://arxiv.org/abs/1304.3931 · HTML: https://mlsanity.com/paper/1304.3931/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 14 Apr 2013.

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

- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)
- [Muon Dynamics as a Spectral Wasserstein Flow](https://mlsanity.com/paper/2604.04891/) (2026)
- [Robust Sublinear Convergence Rates for Iterative Bregman Projections](https://mlsanity.com/paper/2602.01372/) (2026)
- [Quantum Wasserstein GANs for State Preparation at Unseen Points of a Phase Diagram](https://mlsanity.com/paper/2309.09543/) (2023)

## Abstract (arXiv)

We formulate an optimal transport problem for matrix-valued density functions. This is pertinent in the spectral analysis of multivariable time-series. The "mass" represents energy at various frequencies whereas, in addition to a usual transportation cost across frequencies, a cost of rotation is also taken into account. We show that it is natural to seek the transportation plan in the tensor product of the spaces for the two matrix-valued marginals. In contrast to the classical Monge-Kantorovich setting, the transportation plan is no longer supported on a thin zero-measure set.