A RANDOMIZED ALGORITHM FOR PRINCIPAL COMPONENT ANALYSIS

Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a few digits (measured in the spectral norm, relative to the...

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Bibliographic Details
Published inSIAM journal on matrix analysis and applications Vol. 31; no. 3; pp. 1100 - 1124
Main Authors ROKHLIN, Vladimir, SZLAM, Arthur, TYGERT, Mark
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 01.01.2009
Subjects
Accuracy
Algebra
Algorithms
Approximation
Exact sciences and technology
Linear and multilinear algebra, matrix theory
Mathematical analysis
Mathematics
Matrices
Norms
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Principal component analysis
Sciences and techniques of general use
Spectra
Matrix approximation
68W20
low rank
Lanczos
65C60
Algorithm
Numerical analysis
Linear algebra
65F15
Matrices
Algorithm analysis
Power
Principal component analysis
Singular value decomposition
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Summary:Principal component analysis (PCA) requires the computation of a low-rank approximation to a matrix containing the data being analyzed. In many applications of PCA, the best possible accuracy of any rank-deficient approximation is at most a few digits (measured in the spectral norm, relative to the spectral norm of the matrix being approximated). In such circumstances, efficient algorithms have not come with guarantees of good accuracy, unless one or both dimensions of the matrix being approximated are small. We describe an efficient algorithm for the low-rank approximation of matrices that produces accuracy that is very close to the best possible accuracy, for matrices of arbitrary sizes. We illustrate our theoretical results via several numerical examples.
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ISSN:0895-4798
1095-7162
DOI:10.1137/080736417