We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in $\mathcal{P}_2(\Omega)$ 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 metamodelling 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.

翻译：我们在$\mathcal{P}_2(\Omega)$空间中建立了一个以Wasserstein重心为核心，用于稀疏逼近与结构化预测的通用理论及算法框架。该重心的稀疏性体现在：它利用可用测度字典进行计算，但逼近过程仅涉及少量原子。我们证明，稀疏重心类中的最优重构可由我们引入的"最佳n项重心"概念刻画，该概念可视为巴拿赫空间中经典最佳n项逼近的自然推广。研究表明，最佳n项重心是最小化一个高度非凸的双层优化问题的最优解，为此我们发展了实用的数值计算算法策略。进一步，我们利用该逼近工具构建了低计算成本的插值策略，可用于结构化预测及参数化测度族的元建模。通过参数化偏微分方程的模型降阶（MOR）这一具体问题，我们展示了该方法的应用潜力。由于该方法具有稀疏性、自适应性且天然保持质量守恒，它有望克服双曲守恒律中处理间断传播时经典线性方法的已知瓶颈，并为梯度流等测度值偏微分方程的模型降阶开辟新路径。