CAREER: Leveraging Randomization and Structure in Computational Linear Algebra for Data Science

数据科学在解决医疗保健、气候变化和城市规划等社会挑战方面发挥着核心作用。数据科学算法几乎所有发展的核心都是计算线性代数，该领域涉及解决涉及矩阵和其他用于表示数据的线性代数对象的普遍问题的算法的研究。随着数据量的不断增加，随机化已成为开发计算线性代数中高效算法的关键技术。然而，这些算法的理论与实践之间存在巨大差距，这减缓了它们在数据科学应用中的实际采用。该项目确定了关键挑战，并提出了新的方向，以提供必要的算法基础，以确保未来十年在计算数据科学中成功部署广泛的随机线性代数算法。该项目利用理论计算机科学、机器学习、统计学和非线性优化交叉点的基本跨学科思想。除了发展理论基础之外，推动该项目的关键目标之一是促进旨在将随机化纳入 LAPACK 中的持续实施工作，LAPACK 是过去三十年来机器学习、工程、统计和科学计算领域默认的计算线性代数软件包。 该项目的核心是一个综合教育计划，致力于帮助学生获得算法基础和数据科学交叉领域的跨学科技能。 该项目还通过与密歇根大学的工程衔接项目合作，向密歇根州三所资源匮乏的高中的学生进行推广。该项目的目标是缩小理论与实践之间的差距，利用随机化设计改进的算法来解决普遍存在的矩阵问题，例如矩阵乘法、求解线性系统和低阶近似。该项目确定了三个主要目标，即（1）通过黑盒抽样方法重新制定最佳矩阵草图； (2) 通过随机优化的随机迭代细化算法； (3) 研究随机数值线性代数算法的鲁棒性，以保留数据的某些结构元素。 矩阵草图，即输入数据的小型随机近似，是这些算法的关键基础组成部分。该项目旨在开发新的算法和理论方法，以确保矩阵草图和子采样产生的输出的控制和可靠性，这在处理随机化时尤其具有挑战性，并且对于成功的软件集成至关重要。在这些工具的基础上，该项目通过探索确定性迭代求解器和随机优化之间尚未探索的技术，寻求设计解决线性系统和二次问题的高精度算法的新方法。最后，该项目旨在促进对保留数据结构的随机矩阵近似算法的统一理解，这对于特征选择、实验设计、可解释性等至关重要。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

Data science plays a central role in addressing societal challenges, such as healthcare, climate change, and urban planning. At the core of nearly all developments in algorithms for data science is computational linear algebra, an area that concerns the study of algorithms for solving ubiquitous problems involving matrices and other linear-algebraic objects that are used to represent data. With ever-increasing data sizes, randomization has become a key technique for developing efficient algorithms in computational linear algebra. Yet, there is a significant gap between the theory and practice of these algorithms, which has slowed their practical adoption in data science applications. This project identifies key challenges and puts forward new directions towards providing the algorithmic foundations necessary to ensure that a broad scope of randomized linear algebra algorithms are successfully deployed across computational data science over the next decade. This project leverages fundamental interdisciplinary ideas at the intersection of theoretical computer science, machine learning, statistics, and nonlinear optimization. In addition to developing the theoretical foundations, one of the key aims driving the project is to facilitate ongoing implementation efforts aimed at incorporating randomization into LAPACK, the default computational linear algebra software package in machine learning, engineering, statistics, and scientific computing for the past thirty years. At the core of the project is an integrated education plan focused on helping students to gain an interdisciplinary skillset at the intersection of algorithmic foundations and data science. The project also involves outreach to students from three underresourced high schools in Michigan through a collaboration with the university's Engineering Pathways program.The project’s objectives are to close the theory-practice gap in using randomization to design improved algorithms for ubiquitous matrix problems such as matrix multiplication, solving linear systems, and low-rank approximation. The project identifies three major thrusts, namely (1) reformulating optimal matrix sketching via black-box sampling methods; (2) randomized iterative refinement algorithms via stochastic optimization; (3) a study of robustness of randomized numerical linear algebra algorithms to preserve certain structural elements of data. The matrix sketch, i.e., a small randomized approximation of the input data is a key foundational component of these algorithms. The project aims to develop new algorithmic and theoretical approaches towards ensuring the control and reliability of the output produced by matrix sketching and sub-sampling, which is especially challenging when dealing with randomization and will be critical for successful software integration. Building on these tools, the project pursues new approaches for designing high-precision algorithms solving linear systems and quadratic problems, by exploring techniques that lie in the unexplored regime between deterministic iterative solvers and stochastic optimization. Finally, the project aims to contribute to a unified understanding of randomized matrix approximation algorithms that preserve the structure of the data, which is essential for feature selection, experimental design, interpretability, and more.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.