SIAM Review, Volume 62, Issue 3, Page 661-682, January 2020.
Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually suitable for these purposes, but it can be much slower than the unpivoted QR algorithm. For large matrices, the difference in performance is due to increased communication between the processor and slow memory, which QRCP needs in order to choose pivots during decomposition. Our main algorithm, Randomized QR with Column Pivoting (RQRCP), uses randomized projection to make pivot decisions from a much smaller sample matrix, which we can construct to reside in a faster level of memory than the original matrix. This technique may be understood as trading vastly reduced communication for a controlled increase in uncertainty during the decision process. For rank-revealing purposes, the selection mechanism in RQRCP produces results that are the same quality as the standard algorithm, but with performance near that of unpivoted QR (often an order of magnitude faster for large matrices). We also propose two formulas that facilitate further performance improvements. The first efficiently updates sample matrices to avoid computing new randomized projections. The second avoids large trailing updates during the decomposition in truncated low-rank approximations. Our truncated version of RQRCP also provides a key initial step in our truncated SVD approximation, TUXV. These advances open up a new performance domain for large matrix factorizations that will support efficient problem-solving techniques for challenging applications in science, engineering, and data analysis.

中文翻译：

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秩揭示矩阵分解和低秩近似的随机投影

SIAM评论，第62卷，第3期，第661-682页，2020年1月。
秩揭示矩阵分解提供了矩阵频谱分析的重要工具，包括奇异值分解（SVD）和相关的低秩近似技术。带列透视的QR（QRCP）通常适用于这些目的，但是它比未透视的QR算法要慢得多。对于大型矩阵，性能上的差异是由于处理器与慢速存储器之间的通信增加，QRCP要求在分解期间选择枢轴。我们的主要算法是带有列枢轴的随机QR（RQRCP），它使用随机投影来从更小的样本矩阵中做出枢轴决策，我们可以将其构造为比原始矩阵更快地驻留在内存中。可以将这种技术理解为在决策过程中为了控制不确定性而大大减少了通信量。出于排名公开的目的，RQRCP中的选择机制产生的结果与标准算法具有相同的质量，但性能接近未旋转的QR（对于大型矩阵，通常快一个数量级）。我们还提出了两个有助于进一步提高性能的公式。第一种有效地更新样本矩阵，以避免计算新的随机投影。第二个避免了在截断的低秩近似中进行分解时的大量尾随更新。我们的截断版本RQRCP还为我们的截断SVD近似TUXV提供了关键的初始步骤。