Randomized Algorithms for Low-Rank Matrix Factorizations: Sharp Performance Bounds

The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed algorithms in the literature for dimensionality reduction—specific...

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Bibliographic Details
Published in:Algorithmica Vol. 72; no. 1; pp. 264 - 281
Main Authors: Witten, Rafi, Candès, Emmanuel
Format: Journal Article
Language:English
Published: Boston Springer US 01.05.2015
Subjects:
Algorithm Analysis and Problem Complexity
Algorithms
Computer Science
Computer Systems Organization and Communication Networks
Data Structures and Information Theory
Mathematics of Computing
Theory of Computation
Matrix approximation
Dimension reduction
Random matrix
Randomized linear algebra
Pass efficient algorithm
ISSN:0178-4617, 1432-0541
Online Access:Get full text
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Summary:The development of randomized algorithms for numerical linear algebra, e.g. for computing approximate QR and SVD factorizations, has recently become an intense area of research. This paper studies one of the most frequently discussed algorithms in the literature for dimensionality reduction—specifically for approximating an input matrix with a low-rank element. We introduce a novel and rather intuitive analysis of the algorithm in [ 6 ], which allows us to derive sharp estimates and give new insights about its performance. This analysis yields theoretical guarantees about the approximation error and at the same time, ultimate limits of performance (lower bounds) showing that our upper bounds are tight. Numerical experiments complement our study and show the tightness of our predictions compared with empirical observations.
ISSN:0178-4617
1432-0541
DOI:10.1007/s00453-014-9891-7