Wasserstein Optimization in Data Science: Recovery and Sampling

低秩矩阵恢复和采样是当代数据科学中的重要问题，因为它们具有广泛的适用性和与实际场景的直接相关性。值得注意的是，低秩矩阵恢复算法已经被用于分子结构的重建和电子商务的推荐系统，采样算法已经导致系统可以生成艺术、文本和声音。这个项目通过数学优化的视角来解决这些问题。该项目开发的方法利用统一的数学框架，将产生更有效和准确的算法。此外，它开发的理论工具将更广泛地影响优化方法的数学理解。学生将在这个项目中接受培训。该项目开发了利用沃瑟斯坦空间几何来解决矩阵恢复和采样问题的算法。沃瑟斯坦空间的几何结构，或配备了沃瑟斯坦度量的概率测度空间，特别丰富，非常适合于与数据相关的任务。在这个空间中，该项目考虑了1)利用沃瑟斯坦几何变换矩阵传感的优势，以及2)利用梯度流和镜像朗之万方法开发新颖高效的自适应高阶采样算法。在这些组成部分中，新的配方将随着理论论证而发展。特别地，理论论证将展示Wasserstein空间的几何如何比没有适当利用这种几何的现有方法产生真正的实际优势。所提出的方法的应用将包括矩阵补全，相位检索和量子断层扫描，训练神经网络，以及从生成模型中采样。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Low-rank matrix recovery and sampling are essential problems in contemporary data science due to their wide-ranging applicability and direct relevance to practical scenarios. Notably, low-rank matrix recovery algorithms have been used in the reconstruction of molecular structure and recommender systems for e-commerce, and sampling algorithms that have led to systems that can generate art, text, and sound. This project tackles these problems through the lens of mathematical optimization. The methods developed by this project utilize a unified mathematical framework that will yield more efficient and accurate algorithms. Additionally, the theoretical tools it develops will influence the mathematical understanding of optimization methods more broadly. Students will be trained in this project.This project develops algorithms utilizing the geometry of Wasserstein space to solve problems in matrix recovery and sampling. The geometry of Wasserstein space, or the space of probability measures equipped with the Wasserstein metric, is particularly rich and well-suited for data-related tasks. Over this space, the project considers 1) the advantages of utilizing Wasserstein geometry for variations of the matrix sensing, and 2) the utilization of gradient flows and mirror Langevin methods to develop novel and efficient adaptive and higher-order sampling algorithms. In each of these components, novel formulations will be developed along with theoretical justification. In particular, the theoretical justification will show how the geometry of Wasserstein space yields real practical advantages over existing methods that do not properly leverage this geometry. Applications of the proposed methods will include matrix completion, phase retrieval and quantum tomography, training neural networks, and sampling from generative models.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.