We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in \({\mathcal {P}}_2(\varOmega )\) with Wasserstein barycenters. The barycenters are sparse in the sense that they are computed from an available dictionary of measures, but the approximations only involve a reduced number of atoms. We show that the best reconstruction from the class of sparse barycenters is characterized by a notion of best n-term barycenter which we introduce, and which can be understood as a natural extension of the classical concept of best n-term approximation in Banach spaces. We show that the best n-term barycenter is the minimizer of a highly non-convex, bi-level optimization problem, and we develop algorithmic strategies for practical numerical computation. We next leverage this approximation tool to build interpolation strategies that involve a reduced computational cost, and that can be used for structured prediction, and metamodeling of parametrized families of measures. We illustrate the potential of the method through the specific problem of Model Order Reduction (MOR) of parametrized PDEs. Since our approach is sparse, adaptive and preserves mass by construction, it has potential to overcome known bottlenecks of classical linear methods in hyperbolic conservation laws transporting discontinuities. It also paves the way towards MOR for measure-valued PDE problems such as gradient flows.
