Randomized Rank-Revealing QLP for Low-Rank Matrix Decomposition

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.