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.

翻译：枢轴QLP分解通过两次连续的枢轴QR分解计算，可近似奇异值分解。本文研究利用随机化计算低秩矩阵的部分QLP分解方法，称为随机无枢轴QLP（RU-QLP）。与枢轴QLP类似，RU-QLP具有秩揭示能力，但采用随机列采样和无枢轴QR分解。后一项改进使RU-QLP在现代计算平台上具有高度并行化能力。我们对RU-QLP进行分析，在谱范数和Frobenius范数下推导出以下边界：i) 秩揭示性质；ii) 近似子空间与精确奇异子空间及奇异向量之间的主角度；iii) 低秩近似误差。通过数值实验验证了边界的有效性。进一步利用配备GPU的现代多核机器验证RU-QLP的效率。结果表明，与随机SVD相比，RU-QLP在CPU上实现高达7.1倍加速，在GPU上实现高达2.3倍加速。