The Wasserstein barycenter (WB) is an important tool for summarizing sets of probabilities. It finds applications in applied probability, clustering, image processing, etc. When the probability supports are finite, the problem of computing a WB is formulated as a linear optimization problem whose dimensions generally exceed standard solvers' capabilities. For this reason, the WB problem is often replaced with a simpler nonlinear optimization model constructed via an entropic regularization function so that specialized algorithms can be employed to compute an approximate WB efficiently. Contrary to such a widespread inexact scheme, we propose an exact approach based on the Douglas-Rachford splitting method applied directly to the WB linear optimization problem for applications requiring accurate WB.
Our algorithm, which has the interesting interpretation of being built upon averaging marginals, operates a series of simple (and exact) projections that can be parallelized and even randomized, making it suitable for large-scale datasets. As a result, our method achieves good performance in terms of speed while still attaining accuracy. Furthermore, the same algorithm can be applied to compute generalized barycenters of sets of measures with different total masses by allowing for mass creation and destruction upon setting an additional parameter. Numerical comparisons against state-of-the-art methods on several data sets from the literature illustrate the method's performance.
This is a joint work with Daniel Mimouni, Paul Malisani, Jiamin Zhu, from IFP Energies nouvelles.
