Wasserstein Optimization in Data Science: Recovery and Sampling

低阶矩阵恢复和采样是当代数据科学中的基本问题，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，低阶矩阵恢复算法已被用于重建分子结构和电子商务推荐系统，采样算法已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了一个统一的数学框架，将产生更有效和更准确的算法。此外，它开发的理论工具将更广泛地影响对优化方法的数学理解。学生将在这个项目中接受培训。这个项目利用瓦瑟斯坦空间的几何来开发算法，以解决矩阵恢复和采样中的问题。沃瑟斯坦空间的几何，或配备了沃瑟斯坦度量的概率度量空间，特别丰富，非常适合与数据相关的任务。在这个空间中，该项目考虑了1)利用Wasserstein几何进行矩阵传感变化的优点，以及2)利用梯度流和镜像朗之万法来开发新的、高效的自适应和高阶采样算法。在这些成分的每一个中，将开发新的配方并进行理论证明。特别是，理论上的论证将显示出沃瑟斯坦空间的几何如何产生真正的实际优势，而不是现有的方法，这些方法没有适当地利用这种几何。建议方法的应用将包括矩阵补全、相位恢复和量子层析、训练神经网络和从生成模型中采样。该奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.