Randomized algorithms for the low-rank approximation of matrices

• Published in: Proceedings of the National Academy of Sciences - PNAS Vol. 104; no. 51; pp. 20167 - 20172
• Main Authors: Liberty, Edo, Woolfe, Franco, Martinsson, Per-Gunnar, Rokhlin, Vladimir, Tygert, Mark
• Format: Journal Article
• Language: English
• Published: United States National Academy of Sciences 18.12.2007; National Acad Sciences
• Subjects: Absolute value; Algorithms; Approximation; Cost estimates; equations; Integers; Mathematical vectors; Mathematics; Matrices; Physical Sciences; probability; probability analysis; Random variables; Randomized algorithms; Spectral reflectance; Tables of contents
• ISSN: 0027-8424; 1091-6490; 1091-6490
• DOI: 10.1073/pnas.0709640104

We describe two recently proposed randomized algorithms for the construction of low-rank approximations to matrices, and demonstrate their application (inter alia) to the evaluation of the singular value decompositions of numerically low-rank matrices. Being probabilistic, the schemes described here have a finite probability of failure; in most cases, this probability is rather negligible (10⁻¹⁷ is a typical value). In many situations, the new procedures are considerably more efficient and reliable than the classical (deterministic) ones; they also parallelize naturally. We present several numerical examples to illustrate the performance of the schemes.