# Vector and Matrix Optimal Mass Transport: Theory, Algorithm, and Applications

arXiv: https://arxiv.org/abs/1712.10279 · HTML: https://mlsanity.com/paper/1712.10279/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 29 Dec 2017.

## Built on by (most cited first, 3 of 3)

- [Learning quantum data with the quantum Earth Mover's distance](https://mlsanity.com/paper/2101.03037/) (2021)
- [Convergence of Gaussian-smoothed optimal transport distance with sub-gamma distributions and dependent samples](https://mlsanity.com/paper/2103.00394/) (2021)
- [Riemannian block SPD coupling manifold and its application to optimal transport](https://mlsanity.com/paper/2201.12933/) (2022)

## Abstract (arXiv)

In many applications such as color image processing, data has more than one piece of information associated with each spatial coordinate, and in such cases the classical optimal mass transport (OMT) must be generalized to handle vector-valued or matrix-valued densities. In this paper, we discuss the vector and matrix optimal mass transport and present three contributions. We first present a rigorous mathematical formulation for these setups and provide analytical results including existence of solutions and strong duality. Next, we present a simple, scalable, and parallelizable methods to solve the vector and matrix-OMT problems. Finally, we implement the proposed methods on a CUDA GPU and present experiments and applications.