# A novel notion of barycenter for probability distributions based on optimal weak mass transport

arXiv: https://arxiv.org/abs/2102.13380 · HTML: https://mlsanity.com/paper/2102.13380/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 17 of its 54 references that are in the map.
- First posted on arXiv 26 Feb 2021.

## Builds on (most cited first, 17 of 17)

- [Sinkhorn Distances: Lightspeed Computation of Optimal Transportation Distances](https://mlsanity.com/paper/1306.0895/) (2013)
- [Computational Optimal Transport](https://mlsanity.com/paper/1803.00567/) (2018)
- [Fast Computation of Wasserstein Barycenters](https://mlsanity.com/paper/1310.4375/) (2013)
- [Large-Scale Optimal Transport and Mapping Estimation](https://mlsanity.com/paper/1711.02283/) (2017)
- [Robust Optimal Transport with Applications in Generative Modeling and Domain Adaptation](https://mlsanity.com/paper/2010.05862/) (2020)
- [Continuous Regularized Wasserstein Barycenters](https://mlsanity.com/paper/2008.12534/) (2020)
- [Fast Discrete Distribution Clustering Using Wasserstein Barycenter with Sparse Support](https://mlsanity.com/paper/1510.00012/) (2015)
- [Regularized Optimal Transport and the Rot Mover's Distance](https://mlsanity.com/paper/1610.06447/) (2016)
- [Wasserstein barycenters can be computed in polynomial time in fixed dimension](https://mlsanity.com/paper/2006.08012/) (2020)
- [Existence, Duality, and Cyclical monotonicity for weak transport costs](https://mlsanity.com/paper/1809.05893/) (2018)
- [Fr'echet Means and Procrustes Analysis in Wasserstein Space](https://mlsanity.com/paper/1701.06876/) (2017)
- [Parallel Streaming Wasserstein Barycenters](https://mlsanity.com/paper/1705.07443/) (2017)
- [Bayesian Learning with Wasserstein Barycenters](https://mlsanity.com/paper/1805.10833/) (2018)
- [Data-driven regularization of Wasserstein barycenters with an application to multivariate density registration](https://mlsanity.com/paper/1804.08962/) (2018)
- [Stochastic Approximation versus Sample Average Approximation for population Wasserstein barycenters](https://mlsanity.com/paper/2001.07697/) (2020)
- [Wide Consensus for Parallelized Inference](https://mlsanity.com/paper/1511.05350/) (2015)
- [Weak monotone rearrangement on the line](https://mlsanity.com/paper/1902.05763/) (2019)

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

- [Estimating Barycenters of Distributions with Neural Optimal Transport](https://mlsanity.com/paper/2402.03828/) (2024)
- [Energy-Guided Continuous Entropic Barycenter Estimation for General Costs](https://mlsanity.com/paper/2310.01105/) (2023)
- [Tangential Wasserstein Projections](https://mlsanity.com/paper/2207.14727/) (2022)
- [Wasserstein Distributionally Robust Optimization with Wasserstein Barycenters](https://mlsanity.com/paper/2203.12136/) (2022)
- [Approximate Algorithms For k-Sparse Wasserstein Barycenter With Outliers](https://mlsanity.com/paper/2404.13401/) (2024)

## Abstract (arXiv)

We introduce weak barycenters of a family of probability distributions, based on the recently developed notion of optimal weak transport of mass by Gozlanet al. (2017) and Backhoff-Veraguas et al. (2020). We provide a theoretical analysis of this object and discuss its interpretation in the light of convex ordering between probability measures. In particular, we show that, rather than averaging the input distributions in a geometric way (as the Wasserstein barycenter based on classic optimal transport does) weak barycenters extract common geometric information shared by all the input distributions, encoded as a latent random variable that underlies all of them. We also provide an iterative algorithm to compute a weak barycenter for a finite family of input distributions, and a stochastic algorithm that computes them for arbitrary populations of laws. The latter approach is particularly well suited for the streaming setting, i.e., when distributions are observed sequentially. The notion of weak barycenter and our approaches to compute it are illustrated on synthetic examples, validated on 2D real-world data and compared to standard Wasserstein barycenters.