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

秩揭示矩阵分解提供了矩阵谱分析的基本工具，包括奇异值分解 (SVD) 和相关的低秩近似技术。带有列旋转 (QRCP) 的 QR 通常适用于这些目的，但它可能比非旋转 QR 算法慢得多。对于大型矩阵，性能差异是由于处理器和慢速内存之间的通信增加，这是 QRCP 在分解过程中选择主元所需要的。我们的主要算法，带有列旋转的随机 QR (RQRCP)，使用随机投影从一个小得多的样本矩阵中做出枢轴决策，我们可以构建它以驻留在比原始矩阵更快的内存级别。这种技术可以理解为在决策过程中用大大减少的沟通来换取可控的不确定性增加。出于排名显示的目的，RQRCP 中的选择机制产生的结果与标准算法的质量相同，但性能接近非旋转 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。第一个有效地更新样本矩阵以避免计算新的随机投影。第二个避免在截断的低秩近似的分解过程中进行大的尾随更新。我们的 RQRCP 截断版本还为我们的截断 SVD 近似 TUXV 提供了一个关键的初始步骤。