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Singular vectors under random perturbation
Author(s) -
Vu Van
Publication year - 2011
Publication title -
random structures and algorithms
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.314
H-Index - 69
eISSN - 1098-2418
pISSN - 1042-9832
DOI - 10.1002/rsa.20367
Subject(s) - random matrix , perturbation (astronomy) , singular value , mathematics , singular perturbation , struct , multivariate random variable , matrix (chemical analysis) , mathematical analysis , computer science , random variable , eigenvalues and eigenvectors , statistics , physics , quantum mechanics , materials science , composite material , programming language
Computing the first few singular vectors of a large matrix is a problem that frequently comes up in statistics and numerical analysis. Given the presence of noise, an exact calculation is hard to achieve, and the following problem is of importance: How much does a small perturbation to the matrix change the singular vectors? Answering this question, classical theorems, such as those of Davis‐Kahan and Wedin, give tight estimates for the worst‐case scenario. In this paper, we show that if the perturbation (noise) is random and our matrix has low rank, then better estimates can be obtained. Our method relies on high dimensional geometry and is different from those used in earlier papers. © 2011 Wiley Periodicals, Inc. Random Struct. Alg., 2011