# Generalized Wasserstein barycenters between probability measures living on different subspaces

@inproceedings{Delon2021GeneralizedWB, title={Generalized Wasserstein barycenters between probability measures living on different subspaces}, author={Julie Delon and Nathael Gozlan and Alexandre Saint‐Dizier}, year={2021} }

In this paper, we introduce a generalization of the Wasserstein barycenter, to a case where the initial probability measures live on different subspaces of R. We study the existence and uniqueness of this barycenter, we show how it is related to a larger multimarginal optimal transport problem, and we propose a dual formulation. Finally, we explain how to compute numerically this generalized barycenter on discrete distributions, and we propose an explicit solution for Gaussian distributions.

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SHOWING 1-10 OF 28 REFERENCES

Fast Computation of Wasserstein Barycenters

- Computer ScienceICML
- 2014

The Wasserstein distance is proposed to be smoothed with an entropic regularizer and recover in doing so a strictly convex objective whose gradients can be computed for a considerably cheaper computational cost using matrix scaling algorithms.

Discrete Wasserstein barycenters: optimal transport for discrete data

- Computer Science, MathematicsMath. Methods Oper. Res.
- 2016

The results rely heavily on polyhedral theory which is possible due to the discrete structure of the marginals, and it is established that Wasserstein barycenters must also be discrete measures and there is always a barycenter which is provably sparse.

On a construction of multivariate distributions given some multidimensional marginals

- MathematicsAdvances in Applied Probability
- 2019

Abstract In this paper we investigate the link between the joint law of a d-dimensional random vector and the law of some of its multivariate marginals. We introduce and focus on a class of…

Determining point distributions from their projections

- Mathematics2017 International Conference on Sampling Theory and Applications (SampTA)
- 2017

Determining a set of points in ℝd from its projection on lower dimensional spaces is a common task in data analysis. The aim of this note is to overview some general results from the 50s that might…

Iterative Bregman Projections for Regularized Transportation Problems

- MathematicsSIAM J. Sci. Comput.
- 2015

It is shown that for many problems related to optimal transport, the set of linear constraints can be split in an intersection of a few simple constraints, for which the projections can be computed in closed form.

Uniqueness and approximate computation of optimal incomplete transportation plans

- 2011

For α ∈ (0,1) an α-trimming, P ∗, of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according to a positive weight function, f ≤ 1 1−α , in the way…

Optimal maps for the multidimensional Monge-Kantorovich problem

- Mathematics
- 1998

Let μ1,…, μN be Borel probability measures on ℝd. Denote by Γ(μ1,…, μN) the set of all N-tuples T=(T1,…, TN) such that Ti:ℝd ℝd (i = 1,…, N) are Borel-measurable and satisfy μ1 = μi[V] for all Borel…

Uniqueness and approximate computation of optimal incomplete transportation plans

- Mathematics
- 2011

For a given trimming level 2 (0,1) an trimmed version, P , of a probability P is a new probability obtained by re-weighting the probability of any Borel set, B, according a positive weight function,…

Notions of optimal transport theory and how to implement them on a computer

- Computer ScienceComput. Graph.
- 2018

Wasserstein Generative Adversarial Networks

- Computer ScienceICML
- 2017

This work introduces a new algorithm named WGAN, an alternative to traditional GAN training that can improve the stability of learning, get rid of problems like mode collapse, and provide meaningful learning curves useful for debugging and hyperparameter searches.