稀疏Wasserstein重心的近似和结构预报
Approximation and Structured Prediction with Sparse Wasserstein Barycenters
摘要：
我们开发了一个通用的理论和算法框架稀疏近似和结构化预测$\mathcal{P}_2(Ω)$与Wasserstein重心。的重心是稀疏的，在这个意义上，它们是从一个可用的字典的措施，但近似只涉及减少原子的数量计算。我们表明，从稀疏重心类的最佳重建的特点是最好的$n$ -长期重心，我们介绍的概念，这可以理解为一个自然扩张的经典概念的最佳$n$ -长期近似巴拿赫空间。我们表明，最好的$n$长期重心是一个高度非凸，双层最优化问题的最小值，我们开发的算法策略，实际数值计算。接下来，我们利用这个近似工具来构建插值策略，这些策略涉及降低的计算成本，并且可以用于结构化预测和参数化度量族的元建模。我们说明了潜在的方法，通过具体的问题模型降阶（MOR）的参数化偏微分方程。由于我们的方法是稀疏的，自适应的，并保留质量的建设，它有可能克服已知的瓶颈，经典的线性方法在双曲守恒律传输不连续性。它也铺平了道路MOR测量值PDE问题，如梯度流。
英文摘要：
We develop a general theoretical and algorithmic framework for sparse approximation and structured prediction in $\mathcal{P}_2(Ω)$ 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.
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