# Matricial Wasserstein-1 Distance

## TL;DR

This paper introduces a matrix-valued extension of the Wasserstein 1-metric, enabling efficient computation and unbalanced mass transport interpretation for matrix probability densities.

## Contribution

It develops a novel matrix analogue of the Wasserstein 1-metric using duality theory, expanding the applicability of optimal transport to matrix-valued data.

## Key findings

- Provides a duality-based formulation for matrix Wasserstein-1 distance
- Enables easier computation of matrix optimal transport distances
- Offers an unbalanced interpretation of mass transport for matrices

## Abstract

In this note, we propose an extension of the Wasserstein 1-metric ($W_1$) for matrix probability densities, matrix-valued density measures, and an unbalanced interpretation of mass transport. The key is using duality theory, in particular, a "dual of the dual" formulation of $W_1$. This matrix analogue of the Earth Mover's Distance has several attractive features including ease of computation.