A fixed-point approach to barycenters in Wasserstein space

By Pedro C. Álvarez-Esteban, Eustasio del Barrio, Juan Antonio Cuesta-Albertos, Carlos Matrán. 2015

229 citations recorded by the source corpus.

Let $\mathcal{P}_{2,ac}$ be the set of Borel probabilities on $\mathbb{R}^d$ with finite second moment and absolutely continuous with respect to Lebesgue measure. We consider the problem of finding the barycenter (or Fr\'echet mean) of a finite set of probabilities $\nu_1,\ldots,\nu_k \in \mathcal{P}_{2,ac}$ with respect to the $L_2-$Wasserstein metric. For this task we introduce an operator on $\mathcal{P}_{2,ac}$ related to the optimal transport maps pushing forward any $\mu \in \mathcal{P}_{2,ac}$ to $\nu_1,\ldots,\nu_k$. Under very general conditions we prove that the barycenter must be a fixed point for this operator and introduce an iterative procedure which consistently approximates the barycenter. The procedure allows effective computation of barycenters in any location-scatter family, including the Gaussian case. In such cases the barycenter must belong to the family, thus it is characterized by its mean and covariance matrix. While its mean is just the weighted mean of the means of the probabilities, the covariance matrix is characterized in terms of their covariance matrices $\Sigma_1,\dots,\Sigma_k$ through a nonlinear matrix equation. The performance of the iterative procedure in this case is illustrated through numerical simulations, which show fast convergence towards the barycenter.

Subfields: Applied Mathematics, Computational Mechanics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/03373744dc774c1bc96e1003cdf0d462297e902f

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In Applied Mathematics

In Computational Mechanics

In Statistical and Nonlinear Physics

Citations in the corpus

Computational Optimal Transport (2018, background): “13) actually read T (α?) = IRd (the identity map) at the optimal measure α? (the barycenter), and it is shown in [Álvarez-Esteban et al., 2016] that the barycenter α? is the unique (under regularity conditions clarified in [Zemel and Panaretos, 2018, Theo.”
Statistical Aspects of Wasserstein Distances (2018, methodology): “There are local minima that are not global: ´Alvarez-Esteban et al. (2016) construct measures µ 1 , . . . , µ 4 , µ in R 2 such that the average of t µ i µ is the identity, but µ is not the Fr´echet mean.”
Fréchet means and Procrustes analysis in Wasserstein space (2017, background): “Our proof is shorter and arguably simpler, but the proof in [6] shows the additional property that the traces of the matrix iterates are monotonically increasing.”
Automatic Text Evaluation through the Lens of Wasserstein Barycenters (2021, background): “To make it computationally tractable, the measure µ is often constrained to be a discrete measure with free (Cuturi and Doucet, 2014; Álvarez Esteban et al., 2016; Cuturi and Peyré, 2016; Luise et al., 2019) or fixed support (Benamou et al., 2015; Dvurechenskii et al., 2018; Lin et al., 2020;…”