NSF Award
2305315
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.<br/><br/>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.<br/><br/>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.
