AF: Small: Fast and Efficient Randomized Algorithms for Solving Laplacian Systems of Linear Equations and Sparse Least Squares Problems

AF：小型：用于解决线性方程拉普拉斯系统和稀疏最小二乘问题的快速高效随机算法

• 批准号: 1016501
• 负责人: Petros Drineas
• 金额: $ 32.27万
• 依托单位: Rensselaer Polytechnic Institute
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-08-01 至 2015-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1016501&HistoricalAwards=false
• 关键词: AF; Small; Fast; Efficient; Randomized

Randomization in the context of linear-algebraic algorithms is an exciting and innovative idea. In recent years, a large body of work has focused on provably accurate randomized algorithms for regression problems, with a particular emphasis on least-squares regression. Fast algorithms for such problems are of continuous interest due to their broad applicability in scientific computing and statistical data analysis, where increasingly larger input matrices appear. The PI seeks to theoretically and numerically investigate provably accurate and practically useful randomized algorithms for such problems when (i) the constraint matrix of the regression problem is Laplacian, or (ii) the regression problem is under- or over-constrained and sparse. Thus, the PI seeks to address the alarming gap between recent breakthrough theoretical results of Spielman, Teng, and collaborators and their practical applicability, as well as the lack of efficient algorithms dealing with over- or under-constrained regression problems with sparse input matrices. In order to bridge the gap between theory and applications in this line of research, a number of novel theoretical results are necessary and will be investigated. The practical usefulness of the proposed research will be numerically evaluated using data matrices from scientific applications.Efficiently solving large systems of linear equations is perhaps the most fundamental question in numerical analysis and linear algebra, mainly because such systems are ubiquitous in scientific computing applications. The proposed work seeks to bring the theoretical breakthroughs of the recent work of Spielman, Teng, and collaborators on solving systems of linear equations with Laplacian input matrices closer to practice. Towards that end, both theoretical as well as numerical results will be derived. This research paradigm can subsequently be used as a starting point in order to spark further research efforts on broader classes of massive systems of linear equations. A second aspect of the impact of the proposed work has to do with the considerable overlap between Theoretical Computer Science and Numerical Linear Algebra approaches that will be explored. As randomization becomes increasingly useful in the context of linear algebra, the PI expects that the next generation of researchers in this domain will need solid training in both areas, which is exactly what the proposed work will provide to graduate students. Finally, a third aspect of the impact of the proposed work will emerge from the dissemination of our results via workshops, tutorials, and mini-symposia in high-profile relevant conferences.

线性代数算法背景下的随机化是一个令人兴奋且创新的想法。近年来，大量工作集中在可证明准确的回归问题随机算法上，特别强调最小二乘回归。针对此类问题的快速算法因其在科学计算和统计数据分析中的广泛适用性而受到持续关注，其中出现了越来越大的输入矩阵。当 (i) 回归问题的约束矩阵是拉普拉斯矩阵，或 (ii) 回归问题欠约束或过约束且稀疏时，PI 寻求从理论上和数值上研究可证明准确且实际有用的随机算法来解决此类问题。因此，PI 试图解决 Spielman、Teng 及其合作者最近突破性的理论成果与其实际应用性之间的惊人差距，以及缺乏处理稀疏输入矩阵的过约束或欠约束回归问题的有效算法。为了弥合这一研究领域理论与应用之间的差距，需要一些新颖的理论成果，并将对其进行研究。所提出的研究的实际用途将使用科学应用中的数据矩阵进行数值评估。有效求解大型线性方程组可能是数值分析和线性代数中最基本的问题，主要是因为此类系统在科学计算应用中无处不在。这项工作旨在使 Spielman、Teng 及其合作者最近在用拉普拉斯输入矩阵求解线性方程组方面取得的理论突破更接近实践。为此，将得出理论和数值结果。随后可以将该研究范式用作起点，以激发对更广泛类别的大规模线性方程组的进一步研究工作。拟议工作影响的第二个方面与将要探索的理论计算机科学和数值线性代数方法之间的大量重叠有关。随着随机化在线性代数中变得越来越有用，PI 预计该领域的下一代研究人员将需要在这两个领域进行扎实的培训，而这正是拟议的工作将为研究生提供的内容。最后，拟议工作影响的第三个方面将通过研讨会、教程和高调相关会议中的小型研讨会传播我们的成果。