Principal angles between subspaces in an A-based scalar product: Algorithms and perturbation estimates

Computation of principal angles between subspaces is important in many applications, e.g., in statistics and information retrieval. In statistics, the angles are closely related to measures of dependency and covariance of random variables. When applied to column-spaces of matrices, the principal ang...

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Bibliographic Details
Published inSIAM journal on scientific computing Vol. 23; no. 6; pp. 2008 - 2040
Main Authors KNYAZEV, Andrew V, ARGENTATI, Merico E
Format Journal Article
LanguageEnglish
Published Philadelphia, PA Society for Industrial and Applied Mathematics 2002
Subjects
Algorithms
Applied mathematics
Applied sciences
Computer science; control theory; systems
Data processing. List processing. Character string processing
Decomposition
Error analysis
Exact sciences and technology
Information retrieval
Mathematics
Memory organisation. Data processing
Numerical analysis
Numerical analysis. Scientific computation
Numerical linear algebra
Random variables
Sciences and techniques of general use
Software
Software packages
Orthogonal projection
Principal angle
Accuracy
Scalar product
Perturbation analysis
Linear algebra
Rounding error
Subspace
Perturbation
Algorithm
Canonical correlation
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Summary:Computation of principal angles between subspaces is important in many applications, e.g., in statistics and information retrieval. In statistics, the angles are closely related to measures of dependency and covariance of random variables. When applied to column-spaces of matrices, the principal angles describe canonical correlations of a matrix pair. We highlight that all popular software codes for canonical correlations compute only cosine of principal angles, thus making impossible, because of round-off errors, finding small angles accurately. We review a combination of sine and cosine based algorithms that provide accurate results for all angles. We generalize the method to the computation of principal angles in an A-based scalar product for a symmetric and positive definite matrix A. We provide a comprehensive overview of interesting properties of principal angles. We prove basic perturbation theorems for absolute errors for sine and cosine of principal angles with improved constants. Numerical examples and a detailed description of our code are given.
ISSN:1064-8275
1095-7197
DOI:10.1137/S1064827500377332