# Sliced Multi-Marginal Optimal Transport

arXiv: https://arxiv.org/abs/2102.07115 · HTML: https://mlsanity.com/paper/2102.07115/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 28 of its 46 references that are in the map.
- First posted on arXiv 14 Feb 2021.

## Builds on (most cited first, 28 of 28)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [OpenAI Gym](https://mlsanity.com/paper/1606.01540/) (2016)
- [Concrete Problems in AI Safety](https://mlsanity.com/paper/1606.06565/) (2016)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Learning Generative Models with Sinkhorn Divergences](https://mlsanity.com/paper/1706.00292/) (2017)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Generalized Sliced Wasserstein Distances](https://mlsanity.com/paper/1902.00434/) (2019)
- [Gromov-Wasserstein Learning for Graph Matching and Node Embedding](https://mlsanity.com/paper/1901.06003/) (2019)
- [Primal Wasserstein Imitation Learning](https://mlsanity.com/paper/2006.04678/) (2020)
- [Max-Sliced Wasserstein Distance and its use for GANs](https://mlsanity.com/paper/1904.05877/) (2019)
- [Scalable Gromov-Wasserstein Learning for Graph Partitioning and Matching](https://mlsanity.com/paper/1905.07645/) (2019)
- [Statistical and Topological Properties of Sliced Probability Divergences](https://mlsanity.com/paper/2003.05783/) (2020)
- [Distributional Sliced-Wasserstein and Applications to Generative Modeling](https://mlsanity.com/paper/2002.07367/) (2020)
- [Sliced Gromov-Wasserstein](https://mlsanity.com/paper/1905.10124/) (2019)
- [Learning Generative Models across Incomparable Spaces](https://mlsanity.com/paper/1905.05461/) (2019)
- [Scalable Bayes via Barycenter in Wasserstein Space](https://mlsanity.com/paper/1508.05880/) (2015)
- [Multi-marginal optimal transport: theory and applications](https://mlsanity.com/paper/1406.0026/) (2014)
- [Sliced-Wasserstein Autoencoder: An Embarrassingly Simple Generative Model](https://mlsanity.com/paper/1804.01947/) (2018)
- [Orthogonal Estimation of Wasserstein Distances](https://mlsanity.com/paper/1903.03784/) (2019)
- [Multi-marginal Wasserstein GAN](https://mlsanity.com/paper/1911.00888/) (2019)
- [On the Complexity of Approximating Multimarginal Optimal Transport](https://mlsanity.com/paper/1910.00152/) (2019)
- [Wasserstein barycenters are NP-hard to compute](https://mlsanity.com/paper/2101.01100/) (2021)
- [Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm](https://mlsanity.com/paper/1905.13194/) (2019)
- [Wasserstein regularization for sparse multi-task regression](https://mlsanity.com/paper/1805.07833/) (2018)
- [Estimating Barycenters of Measures in High Dimensions](https://mlsanity.com/paper/2007.07105/) (2020)
- [Polynomial-time algorithms for Multimarginal Optimal Transport problems with structure](https://mlsanity.com/paper/2008.03006/) (2020)
- [Hardness results for Multimarginal Optimal Transport problems](https://mlsanity.com/paper/2012.05398/) (2020)
- [Multimarginal Optimal Transport by Accelerated Alternating Minimization](https://mlsanity.com/paper/2004.02294/) (2020)

## Built on by (most cited first, 5 of 5)

- [Fast Approximation of the Sliced-Wasserstein Distance Using Concentration of Random Projections](https://mlsanity.com/paper/2106.15427/) (2021)
- [Faster Unbalanced Optimal Transport: Translation invariant Sinkhorn and 1-D Frank-Wolfe](https://mlsanity.com/paper/2201.00730/) (2022)
- [Factored couplings in multi-marginal optimal transport via difference of convex programming](https://mlsanity.com/paper/2110.00629/) (2021)
- [On Combining Expert Demonstrations in Imitation Learning via Optimal Transport](https://mlsanity.com/paper/2307.10810/) (2023)
- [Leveraging Optimal Transport via Projections on Subspaces for Machine Learning Applications](https://mlsanity.com/paper/2311.13883/) (2023)

## Abstract (arXiv)

Multi-marginal optimal transport enables one to compare multiple probability measures, which increasingly finds application in multi-task learning problems. One practical limitation of multi-marginal transport is computational scalability in the number of measures, samples and dimensionality. In this work, we propose a multi-marginal optimal transport paradigm based on random one-dimensional projections, whose (generalized) distance we term the sliced multi-marginal Wasserstein distance. To construct this distance, we introduce a characterization of the one-dimensional multi-marginal Kantorovich problem and use it to highlight a number of properties of the sliced multi-marginal Wasserstein distance. In particular, we show that (i) the sliced multi-marginal Wasserstein distance is a (generalized) metric that induces the same topology as the standard Wasserstein distance, (ii) it admits a dimension-free sample complexity, (iii) it is tightly connected with the problem of barycentric averaging under the sliced-Wasserstein metric. We conclude by illustrating the sliced multi-marginal Wasserstein on multi-task density estimation and multi-dynamics reinforcement learning problems.