Fast Parallel Randomized QR with Column Pivoting Algorithms for Reliable Low-rank Matrix Approximations

Factorizing large matrices by QR with column pivoting (QRCP) is substantially more expensive than QR without pivoting, owing to communication costs required for pivoting decisions. In contrast, randomized QRCP (RQRCP) algorithms have proven themselves empirically to be highly competitive with high-performance implementations of QR in processing time, on uniprocessor and shared memory machines, and as reliable as QRCP in pivot quality. We show that RQRCP algorithms can be as reliable as QRCP with failure probabilities exponentially decaying in oversampling size. We also analyze efficiency differences among different RQRCP algorithms. More importantly, we develop distributed memory implementations of RQRCP that are significantly better than QRCP implementations in ScaLAPACK. As a further development, we introduce the concept of and develop algorithms for computing spectrum-revealing QR factorizations for low-rank matrix approximations, and demonstrate their effectiveness against leading low-rank approximation methods in both theoretical and numerical reliability and efficiency.Comment: 11 pages, 14 figures, accepted by 2017 IEEE 24th International Conference on High Performance Computing (HiPC), awarded the best paper priz

Householder QR Factorization With Randomization for Column Pivoting (HQRRP)

A fundamental problem when adding column pivoting to the Householder QR fac- torization is that only about half of the computation can be cast in terms of high performing matrix- matrix multiplications, which greatly limits the bene ts that can be derived from so-called blocking of algorithms. This paper describes a technique for selecting groups of pivot vectors by means of randomized projections. It is demonstrated that the asymptotic op count for the proposed method is 2mn2 �����(2=3)n3 for an m n matrix, identical to that of the best classical unblocked Householder QR factorization algorithm (with or without pivoting). Experiments demonstrate acceleration in speed of close to an order of magnitude relative to the geqp3 function in LAPACK, when executed on a modern CPU with multiple cores. Further, experiments demonstrate that the quality of the randomized pivot selection strategy is roughly the same as that of classical column pivoting. The described algorithm is made available under Open Source license and can be used with LAPACK or libflame