393 research outputs found
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
Low-Rank Matrix Approximations with Flip-Flop Spectrum-Revealing QR Factorization
We present Flip-Flop Spectrum-Revealing QR (Flip-Flop SRQR) factorization, a
significantly faster and more reliable variant of the QLP factorization of
Stewart, for low-rank matrix approximations. Flip-Flop SRQR uses SRQR
factorization to initialize a partial column pivoted QR factorization and then
compute a partial LQ factorization. As observed by Stewart in his original QLP
work, Flip-Flop SRQR tracks the exact singular values with "considerable
fidelity". We develop singular value lower bounds and residual error upper
bounds for Flip-Flop SRQR factorization. In situations where singular values of
the input matrix decay relatively quickly, the low-rank approximation computed
by SRQR is guaranteed to be as accurate as truncated SVD. We also perform a
complexity analysis to show that for the same accuracy, Flip-Flop SRQR is
faster than randomized subspace iteration for approximating the SVD, the
standard method used in Matlab tensor toolbox. We also compare Flip-Flop SRQR
with alternatives on two applications, tensor approximation and nuclear norm
minimization, to demonstrate its efficiency and effectiveness
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