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:我们为瓦塞斯坦酒保中心开发了一个一般的理论和算法框架,用于与瓦塞斯坦酒保中心一起对少许近似和结构化预测,用瓦塞斯坦酒保中心开发出一个一般的理论和算法框架。 酒保中心人很少, 因为他们是从现有计量词典中计算出来的, 但近似只涉及较少的原子数量。 我们显示,从少许酒保中心组类中进行的最佳重建的特点是我们引入一个最佳的美元定期调价概念, 并且可以理解为Banach空间中最优美元中期近似传统概念的自然延伸。 我们表明,最好的美元期价调价中心是尽可能减少高度非消费、双级优化问题的最小化,我们为实际数字计算制定了算法战略。 我们接下来利用这一近似工具来建立涉及降低计算成本的内推法, 可用于结构化的预测, 并模拟各种计量措施的组合。 我们用这个方法通过模型修饰(MOR) 减少价格规则(MARM) 来减少高度流, 也通过已知的平质性运输方法来保持这种变压压。