# Unnormalized Optimal Transport

arXiv: https://arxiv.org/abs/1902.03367 · HTML: https://mlsanity.com/paper/1902.03367/

## Key facts

- Cited by 15 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 9 Feb 2019.

## Built on by (most cited first, 15 of 15)

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- [Sliced Optimal Partial Transport](https://mlsanity.com/paper/2212.08049/) (2022)
- [Entropy Partial Transport with Tree Metrics: Theory and Practice](https://mlsanity.com/paper/2101.09756/) (2021)
- [Learning High Dimensional Wasserstein Geodesics](https://mlsanity.com/paper/2102.02992/) (2021)
- [Scalable Unbalanced Sobolev Transport for Measures on a Graph](https://mlsanity.com/paper/2302.12498/) (2023)
- [The Signed Cumulative Distribution Transform for 1-D Signal Analysis and Classification](https://mlsanity.com/paper/2106.02146/) (2021)
- [Integrating Dynamical Systems Modeling with Spatiotemporal scRNA-seq Data Analysis](https://mlsanity.com/paper/2503.11347/) (2025)
- [Robot Policy Learning with Temporal Optimal Transport Reward](https://mlsanity.com/paper/2410.21795/) (2024)
- [A scalable deep learning approach for solving high-dimensional dynamic optimal transport](https://mlsanity.com/paper/2205.07521/) (2022)
- [A geometric perspective on functional outlier detection](https://mlsanity.com/paper/2109.06849/) (2021)
- [PTLp: Partial Transport Lp Distances](https://mlsanity.com/paper/2307.13571/) (2023)
- [Accelerated nonlinear primal-dual hybrid gradient methods with applications to supervised machine learning](https://mlsanity.com/paper/2109.12222/) (2021)
- [Optimal Transport Features for Morphometric Population Analysis](https://mlsanity.com/paper/2208.05891/) (2022)
- [An Efficient Orlicz-Sobolev Approach for Transporting Unbalanced Measures on a Graph](https://mlsanity.com/paper/2502.00739/) (2025)
- [Neural solver for Wasserstein Geodesics and optimal transport dynamics](https://mlsanity.com/paper/2602.22003/) (2026)

## Abstract (arXiv)

We propose an extension of the computational fluid mechanics approach to the Monge-Kantorovich mass transfer problem, which was developed by Benamou-Brenier. Our extension allows optimal transfer of unnormalized and unequal masses. We obtain a one-parameter family of simple modifications of the formulation in [4]. This leads us to a new Monge-Ampere type equation and a new Kantorovich duality formula. These can be solved efficiently by, for example, the Chambolle-Pock primal-dual algorithm. This solution to the extended mass transfer problem gives us a simple metric for computing the distance between two unnormalized densities. The L1 version of this metric was shown in [23] (which is a precursor of our work here) to have desirable properties.