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

在线性代数算法的背景下，随机化是一个令人兴奋和创新的想法。近年来，大量的工作集中在回归问题的可证明准确的随机化算法上，尤其是最小二乘回归。这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一直受到人们的关注，在这些领域中，输入矩阵越来越大。当(I)回归问题的约束矩阵是拉普拉斯的，或者(Ii)回归问题是欠约束或过约束的稀疏问题时，PI试图从理论和数值上研究这类问题的可证明的准确和实用的随机算法。因此，PI试图解决Spielman、Teng和合作者最近突破性的理论结果和它们的实际适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的理论和应用之间的差距，一些新的理论结果是必要的，并将被调查。有效地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，主要是因为这类系统在科学计算应用中是普遍存在的。这项拟议的工作旨在使Spielman、Teng和合作者最近在求解具有拉普拉斯输入矩阵的线性方程组方面的理论突破更接近于实践。为此，将得到理论和数值结果。这一研究范式随后可以作为一个起点，以激发对更广泛类别的大规模线性方程组的进一步研究努力。拟议工作的影响的第二个方面涉及理论计算机科学和将被探索的数值线性代数方法之间的相当大的重叠。随机化在线性代数的背景下变得越来越有用，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，这正是拟议的工作将提供给研究生的。最后，拟议工作的第三个影响方面将通过研讨会、教程和在相关会议上举行的小型研讨会传播我们的成果。

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.