A Smoothed Dual Approach for Variational Wasserstein Problems

Variational problems that involve Wasserstein distances have been recently proposed to summarize and learn from probability measures. Despite being conceptually simple, such problems are computationally challenging because they involve minimizing over quantities (Wasserstein distances) that are themselves hard to compute. We show that the dual formulation of Wasserstein variational problems introduced recently by Carlier et al. (2014) can be regularized using an entropic smoothing, which leads to smooth, differentiable, convex optimization problems that are simpler to implement and numerically more stable. We illustrate the versatility of this approach by applying it to the computation of Wasserstein barycenters and gradient flows of spacial regularization functionals

Distributed optimization with quantization for computing Wasserstein barycenters

We study the problem of the decentralized computation of entropy-regularized semi-discrete Wasserstein barycenters over a network. Building upon recent primal-dual approaches, we propose a sampling gradient quantization scheme that allows efficient communication and computation of approximate barycenters where the factor distributions are stored distributedly on arbitrary networks. The communication and algorithmic complexity of the proposed algorithm are shown, with explicit dependency on the size of the support, the number of distributions, and the desired accuracy. Numerical results validate our algorithmic analysis

Solving general elliptical mixture models through an approximate Wasserstein manifold

We address the estimation problem for general finite mixture models, with a particular focus on the elliptical mixture models (EMMs). Compared to the widely adopted Kullback-Leibler divergence, we show that the Wasserstein distance provides a more desirable optimisation space. We thus provide a stable solution to the EMMs that is both robust to initialisations and reaches a superior optimum by adaptively optimising along a manifold of an approximate Wasserstein distance. To this end, we first provide a unifying account of computable and identifiable EMMs, which serves as a basis to rigorously address the underpinning optimisation problem. Due to a probability constraint, solving this problem is extremely cumbersome and unstable, especially under the Wasserstein distance. To relieve this issue, we introduce an efficient optimisation method on a statistical manifold defined under an approximate Wasserstein distance, which allows for explicit metrics and computable operations, thus significantly stabilising and improving the EMM estimation. We further propose an adaptive method to accelerate the convergence. Experimental results demonstrate the excellent performance of the proposed EMM solver.Comment: This work has been accepted to AAAI2020. Note that this version also corrects a small error on the Equation (16) in proo