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-p...

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Veröffentlicht in:Proceedings, 24th IEEE International Conference on High Performance Computing : 18-21 December 2017, Jaipur, India S. 233 - 242
Hauptverfasser: Xiao, Jianwei, Gu, Ming, Langou, Julien
Format: Tagungsbericht
Sprache:Englisch
Veröffentlicht: IEEE 01.12.2017
Schlagworte:
Algorithm design and analysis
Approximation algorithms
Decorrelation
distributed computing
low-rank approximation
Matrix decomposition
QR factorization
randomization spectrum-revealing
Reliability theory
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Zusammenfassung: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.
DOI:10.1109/HiPC.2017.00035