Approximate solutions to large least squares problems can be computed efficiently using leverage score-based row-sketches, but directly computing the leverage scores, or sampling according to them with naive methods, still requires an expensive manipulation and processing of the design matrix. In this paper we develop efficient leverage score-based sampling methods for matrices with certain Kronecker product-type structure; in particular we consider matrices that are monotone lower column subsets of Kronecker product matrices. Our discussion is general, encompassing least squares problems on infinite domains, in which case matrices formally have infinitely many rows. We briefly survey leverage score-based sampling guarantees from the numerical linear algebra and approximation theory communities, and follow this with efficient algorithms for sampling when the design matrix has Kronecker-type structure. Our numerical examples confirm that sketches based on exact leverage score sampling for our class of structured matrices achieve superior residual compared to approximate leverage score sampling methods.

翻译：使用基于杠杆分数的行针,可以高效率地计算出解决大最小方问题的近似办法,但直接计算杠杆分数,或按其采用天真的方法进行抽样,这仍然需要对设计矩阵进行昂贵的操纵和处理。在本文件中,我们为具有某些克罗内克产品类型结构的矩阵开发了高效的基于杠杆分数的抽样方法;特别是,我们认为矩阵是克罗内克产品矩阵的单体下列子。我们的讨论是一般性的,包括无限域的最小方数问题,在这种情况下,矩阵形式上是无限多行的。我们简要调查数字线性代数和近似理论界基于杠杆的抽样保证,并在设计矩阵有克伦内克型结构时采用高效的抽样算法。我们的数字实例证实,基于我们各类结构矩阵的精确率分数抽样的草图与近似杠杆分数的抽样方法相比,具有较高的剩余值。