Doubly Regularized Entropic Wasserstein Barycenters
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We study a general formulation of regularized Wasserstein barycenters that enjoys favorable regularity, approximation, stability and (grid-free) optimization properties. This barycenter is defined as the unique probability measure that minimizes the sum of entropic optimal transport (EOT) costs with respect to a family of given probability measures, plus an entropy term. We denote it (λ,τ)-barycenter, where λ is the inner regularization strength and τ the outer one. This formulation recovers several previously proposed EOT barycenters for various choices of λ,τ≥ 0 and generalizes them. First, in spite of – and in fact owing to – being doubly regularized, we show that our formulation is debiased for τ=λ/2: the suboptimality in the (unregularized) Wasserstein barycenter objective is, for smooth densities, of the order of the strength λ^2 of entropic regularization, instead of max{λ,τ} in general. We discuss this phenomenon for isotropic Gaussians where all (λ,τ)-barycenters have closed form. Second, we show that for λ,τ>0, this barycenter has a smooth density and is strongly stable under perturbation of the marginals. In particular, it can be estimated efficiently: given n samples from each of the probability measures, it converges in relative entropy to the population barycenter at a rate n^-1/2. And finally, this formulation lends itself naturally to a grid-free optimization algorithm: we propose a simple noisy particle gradient descent which, in the mean-field limit, converges globally at an exponential rate to the barycenter.
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