SIAM Review, Volume 62, Issue 3, Page 659-659, January 2020.
The SIGEST article in this issue, “Randomized Projection for Rank-Revealing Matrix Factorizations and Low-Rank Approximations,” by Jed A. Duersch and Ming Gu, concerns the use of randomization to reduce the bottleneck in a central numerical linear algebra kernel. The authors consider QR factorization, which is a key component in algorithms for constructing stable bases, for solving least-squares problems, and generally for summarizing or extracting insights from data. Column pivoting is the standard technique for dealing with cases where the target matrix is not of full rank, or where the rank is in doubt. However, for large-scale problems column pivoting requires substantial communication between different levels of computer memory, which severely limits the overall efficiency. The authors therefore propose a strategy for choosing pivots based on randomization rather than exhaustive search. This approach reduces communication complexity while increasing uncertainty in the quality of the factorization in a controlled manner. The article makes use of clear pseudocode listings in order to summarize both the original deterministic strategies and the newly developed randomized approaches; see Algorithms 1 to 6. The authors also explain how a Bayesian framework can be set up to quantify the uncertainty introduced by the use of random projections. Extensive, large-scale tests are conducted using Fortran with OpenMP, an application programming interface that supports shared-memory multiprocessing. The original article appeared in the SIAM Journal on Scientific Computing in 2017, and the main idea has subsequently proved to be effective for a number of related tasks. In creating this highlighted article for the SIGEST section, the authors have revised the introductory material, providing additional context; reorganized the document structure and added new visualizations; updated and extended the computational experiments, while making this section more compact and easier to interpret; and discussed recent developments in the field and provided references to the growing literature.

中文翻译：

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SIGEST

SIAM评论，第62卷，第3期，第659-659页，2020年1月。
Jed A. Duersch和Ming Gu撰写的本期SIGEST文章“用于秩揭示矩阵因式分解和低秩逼近的随机投影”涉及使用随机化来减少中央数字线性代数核的瓶颈。作者考虑了QR分解，QR分解是构建稳定基础，解决最小二乘问题以及通常用于汇总或从数据中提取见解的算法中的关键组成部分。列透视是处理目标矩阵不完全排名或排名不确定的情况的标准技术。但是，对于大规模问题，列透视图需要在不同级别的计算机内存之间进行大量通信，这严重限制了整体效率。因此，作者提出了一种基于随机而非穷举搜索选择枢轴的策略。这种方法降低了通信复杂性，同时以受控方式增加了因式分解质量的不确定性。本文利用清晰的伪代码清单来总结原始的确定性策略和新开发的随机方法。参见算法1至6。作者还解释了如何建立贝叶斯框架来量化使用随机投影引入的不确定性。使用Fortran和OpenMP（支持共享内存多处理的应用程序编程接口）进行了广泛的大规模测试。原始文章发表在2017年的《 SIAM科学计算期刊》上，其主要思想随后被证明对许多相关任务有效。在为SIGEST部分创建此突出显示的文章时，作者对介绍性材料进行了修改，从而提供了更多的上下文。重组了文档结构并添加了新的可视化；更新并扩展了计算实验，同时使本节更紧凑，更易于解释；并讨论了该领域的最新发展，并为不断增长的文献提供了参考。同时使本节更紧凑，更易于解释；并讨论了该领域的最新发展，并为不断增长的文献提供了参考。同时使本节更紧凑，更易于解释；并讨论了该领域的最新发展，并为不断增长的文献提供了参考。