Tracking a few extreme singular values and vectors in signal processing

In various applications it is necessary to keep track of a low-rank approximation of a covariance matrix, R(t), slowly varying with time. It is convenient to track the left singular vectors associated with the largest singular values of the triangular factor, L(t), of its Cholesky factorization. The...

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Bibliographic Details
Published in:Proceedings of the IEEE 1990-08, Vol.78 (8), p.1327-1343
Main Authors: Comon, P., Golub, G.H.
Format: Article
Language:English
Subjects:
Applied sciences
Array signal processing
Computer Science
Covariance matrix
Eigenvalues and eigenfunctions
Exact sciences and technology
Fuzzy sets
Information, signal and communications theory
Least squares approximation
Mathematical methods
Signal and Image Processing
Signal processing
Signal processing algorithms
Singular value decomposition
Sparse matrices
Symmetric matrices
Telecommunications and information theory
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Summary:In various applications it is necessary to keep track of a low-rank approximation of a covariance matrix, R(t), slowly varying with time. It is convenient to track the left singular vectors associated with the largest singular values of the triangular factor, L(t), of its Cholesky factorization. These algorithms are referred to as square-root. The drawback of the eigenvalue decomposition (EVD) or the singular value decompositions (SVD) is usually the volume of the computations. Various numerical methods for carrying out this task are surveyed, and it is shown why this heavy computational burden is questionable in numerous situations and should be revised. Indeed, the complexity per eigenpair is generally a quadratic function of the problem size, but there exist faster algorithms with linear complexity. Finally, in order to make a choice among the large and fuzzy set of available techniques, comparisons based on computer simulations in a relevant signal processing context are made.< >
ISSN:0018-9219
1558-2256
DOI:10.1109/5.58320