Barycenters in the Wasserstein Space

By Martial Agueh, Guillaume Carlier. 2011

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In Applied Mathematics

Citations in the corpus

Federated Learning with Matched Averaging (2020, methodology): “In a special case where all local neural networks and the global model are assumed to have same number of hidden neurons, solving equation 2 is equivalent to finding a Wasserstein barycenter (Agueh & Carlier, 2011) of the empirical distributions over the weights of local neural networks.”
Model Fusion via Optimal Transport (2019, methodology): “The resulting layer-wise averaging scheme can be interpreted as computing the Wasserstein barycenter [3, 4] of the probability measures defined at the corresponding layers of the parent models.”
Wasserstein Fair Classification (2019, background): “Let T ( p ∗ ) be the set of transportation maps with elements Therefore we are interested in which coincides with the Wasserstein-1 barycenter with normalized subgroup size as weight to every group distribution p S a (Agueh and Carlier, 2011).”
Computational Optimal Transport (2018, background): “As shown in [Agueh and Carlier, 2011], first order optimality conditions of the barycenter problem (9.”
Theoretical Analysis of Domain Adaptation with Optimal Transport (2016, background): “Indeed, as shown in [1], for a quadratic Euclidean cost function the solution μN of the barycenter problem in the Wasserstein space is given by the following equation:”
Iterative Bregman Projections for Regularized Transportation Problems (2014, methodology): “OT barycenters ﬁnd applications for instance in statistics to deﬁne a mean empirical estimator from a family of observed histograms [10], or in machine learning [27] to provide an extended deﬁnition of k -means clustering and compute average histograms-of-features under the OT metric.”
Wasserstein Barycenter and Its Application to Texture Mixing (2011, neutral): “Independently to our work, Agueh and Carlier have performed a mathematical analysis of this problem [30].”
Entropic optimal transport: convergence of potentials (2021, methodology): “The resulting “multimarginal” optimal transport problem has become a focus of attention as the primary tool to analyze Wasserstein barycenters in the sense of [1].”
Scaling Algorithms for Unbalanced Transport Problems (2016, background): “6) is a formulation of the Fréchet means in the Wasserstein space, also known as Wasserstein barycenters (see [1]).”
Sinkhorn Barycenters with Free Support via Frank-Wolfe Algorithm (2019, background): “Since the (Wasserstein) barycenter of Gaussian distributions can be estimated accurately (see [2]), in Fig.”
On the Bures–Wasserstein distance between positive definite matrices (2017, methodology): “We have proved the following theorem first obtained in [1], building upon the earlier work in [22] and [31]”
The Wasserstein space of stochastic processes (2021, background): “Famously, Agueh and Carlier [3] introduced the concept of barycenters w.”
Context Mover's Distance & Barycenters: Optimal transport of contexts for building representations (2018, background): “It has been shown in Agueh & Carlier (2011) that when the underlying space is Euclidean and p = 2 , there exists a unique minimizer to the Wasserstein barycenter problem.”
Inference for empirical Wasserstein distances on finite spaces (2016, methodology): “The barycentre with respect to the Wasserstein metric (Agueh and Carlier, 2011) has been shown to elicit important structure from complex data and to be a promising tool, e.g. in deformable models (Boissard et al., 2015; Agulló-Antolı́n et al., 2015). It has also been used in large-scale Bayesian in”
Bayesian Learning with Wasserstein Barycenters (2018, methodology): “2) can be relaxed to the condition Γ ({m : m λ}) > 0, as shown in [1] or [25, Theorem 5.”
Sliced Wasserstein Distance for Learning Gaussian Mixture Models (2017, methodology): “Above minimization is known as the Wasserstein means problem and is closely related to the Wasserstein Barycenter problem [1, 45, 13, 20].”
Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling (2015, background): “7) has been first introduced in [1], where the authors also propose an interpretation in terms of a multi-marginal optimal transport problem.”