# Unbalanced Optimal Transport: Dynamic and Kantorovich Formulation

arXiv: https://arxiv.org/abs/1508.05216 · HTML: https://mlsanity.com/paper/1508.05216/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Aug 2015.

## Built on by (most cited first, 6 of 6)

- [Tree Mover's Distance: Bridging Graph Metrics and Stability of Graph Neural Networks](https://mlsanity.com/paper/2210.01906/) (2022)
- [Unbalancedness in Neural Monge Maps Improves Unpaired Domain Translation](https://mlsanity.com/paper/2311.15100/) (2023)
- [GENOT: Entropic (Gromov) Wasserstein Flow Matching with Applications to Single-Cell Genomics](https://mlsanity.com/paper/2310.09254/) (2023)
- [The Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification](https://mlsanity.com/paper/2106.02146/) (2021)
- [Algorithms for Weak Optimal Transport with an Application to Economics](https://mlsanity.com/paper/2205.09825/) (2022)
- [An Homogeneous Unbalanced Regularized Optimal Transport model with applications to Optimal Transport with Boundary](https://mlsanity.com/paper/2201.02082/) (2022)

## Abstract (arXiv)

This article presents a new class of distances between arbitrary nonnegative Radon measures inspired by optimal transport. These distances are defined by two equivalent alternative formulations: (i) a dynamic formulation defining the distance as a geodesic distance over the space of measures (ii) a static "Kantorovich" formulation where the distance is the minimum of an optimization problem over pairs of couplings describing the transfer (transport, creation and destruction) of mass between two measures. Both formulations are convex optimization problems, and the ability to switch from one to the other depending on the targeted application is a crucial property of our models. Of particular interest is the Wasserstein-Fisher-Rao metric recently introduced independently by Chizat et al. and Kondratyev et al. Defined initially through a dynamic formulation, it belongs to this class of metrics and hence automatically benefits from a static Kantorovich formulation.