Title: Randomized Projection for Rank-Revealing Matrix Factorizations and Low-Rank Approximations

Abstract

Rank-revealing matrix decompositions provide an essential tool in spectral analysis of matrices, including the Singular Value Decomposition (SVD) and related low-rank approximation techniques. QR with Column Pivoting (QRCP) is usually suitable for these purposes, but it can be much slower than the unpivoted QR algorithm. For large matrices, the difference in performance is due to increased communication between the processor and slow memory, which QRCP needs in order to choose pivots during decomposition. Our main algorithm, Randomized QR with Column Pivoting (RQRCP), uses randomized projection to make pivot decisions from a much smaller sample matrix, which we can construct to reside in a faster level of memory than the original matrix. This technique may be understood as trading vastly reduced communication for a controlled increase in uncertainty during the decision process. Furthermore, for rank-revealing purposes, the selection mechanism in RQRCP produces results that are the same quality as the standard algorithm, but with performance near that of unpivoted QR (often an order of magnitude faster for large matrices). Additionally, we also propose two formulas that facilitate further performance improvements. The first efficiently updates sample matrices to avoid computing new randomized projections. The second avoids large trailing updates during the decompositionmore » in truncated low-rank approximations. Our truncated version of RQRCP also provides a key initial step in our truncated SVD approximation, TUXV. These advances open up a new performance domain for large matrix factorizations that will support efficient problem-solving techniques for challenging applications in science, engineering, and data analysis.« less

Authors:

Duersch, Jed A. ^[1]; Gu, Ming ^[2]

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+ Show Author Affiliations

1. Sandia National Lab. (SNL-CA), Livermore, CA (United States)
2. Univ. of California, Berkeley, CA (United States)

Publication Date:

Thu Aug 06 00:00:00 EDT 2020

Research Org.:

Sandia National Lab. (SNL-CA), Livermore, CA (United States)

Sponsoring Org.:

USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)

OSTI Identifier:

1650159

Report Number(s):

SAND-2020-8281J
Journal ID: ISSN 0036-1445; 689858

Grant/Contract Number:  

AC04-94AL85000; NA0003525; 1319312; 1760316

Resource Type:

Accepted Manuscript

Journal Name:

SIAM Review

Additional Journal Information:

Journal Volume: 62; Journal Issue: 3; Journal ID: ISSN 0036-1445

Publisher:

Society for Industrial and Applied Mathematics

Country of Publication:

United States

Language:

English

Subject:

97 MATHEMATICS AND COMPUTING; QR factorization; column pivoting; rank-revealing; random sampling; sample update; blocked algorithm; low-rank approximation; truncated SVD