The pivoted QLP decomposition is computed through two consecutive pivoted QR decompositions, and provides an approximation to the singular value decomposition. This work is concerned with a partial QLP decomposition of low-rank matrices computed through randomization, termed Randomized Unpivoted QLP (RU-QLP). Like pivoted QLP, RU-QLP is rank-revealing and yet it utilizes random column sampling and the unpivoted QR decomposition. The latter modifications allow RU-QLP to be highly parallelizable on modern computational platforms. We provide an analysis for RU-QLP, deriving bounds in spectral and Frobenius norms on: i) the rank-revealing property; ii) principal angles between approximate subspaces and exact singular subspaces and vectors; and iii) low-rank approximation errors. Effectiveness of the bounds is illustrated through numerical tests. We further use a modern, multicore machine equipped with a GPU to demonstrate the efficiency of RU-QLP. Our results show that compared to the randomized SVD, RU-QLP achieves a speedup of up to 7.1 times on the CPU and up to 2.3 times with the GPU.

中文翻译：

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用于低秩矩阵分解的随机秩揭示 QLP

枢轴 QLP 分解是通过两个连续的枢轴 QR 分解计算的，并提供奇异值分解的近似值。这项工作涉及通过随机化计算的低秩矩阵的部分 QLP 分解，称为 Randomized Unpivoted QLP (RU-QLP)。与枢轴 QLP 一样，RU-QLP 是排名显示的，但它利用随机列采样和非枢轴 QR 分解。后者的修改允许 RU-QLP 在现代计算平台上高度并行化。我们为 RU-QLP 提供了分析，在以下方面推导了光谱和 Frobenius 范数的界限： i) 秩揭示属性；ii) 近似子空间与精确奇异子空间和向量之间的主角；iii) 低秩近似误差。界限的有效性通过数值测试来说明。我们进一步使用配备 GPU 的现代多核机器来展示 RU-QLP 的效率。我们的结果表明，与随机 SVD 相比，RU-QLP 在 CPU 上实现了高达 7.1 倍的加速，在 GPU 上实现了高达 2.3 倍的加速。