Quadratic optimization for simultaneous matrix diagonalization

Simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics. Although objective functions in a least-squares sense can be easily formulated, their minimization is not trivial, because constraints and fou...

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Bibliographic Details
Published in:IEEE transactions on signal processing 2006-09, Vol.54 (9), p.3270-3278
Main Authors: Vollgraf, R., Obermayer, K.
Format: Article
Language:English
Subjects:
Algorithms
Applied sciences
Blind source separation
Blind source separation (BSS)
Constrictions
Coordinate measuring machines
Detection, estimation, filtering, equalization, prediction
Eigenvalues and eigenfunctions
Exact sciences and technology
Information, signal and communications theory
Jacobian matrices
joint diagonalization
Mathematical analysis
Matrices
Matrix methods
Miscellaneous
Optimization
Optimization algorithms
Principal component analysis
quadratic optimization
Signal and communications theory
Signal processing
Signal processing algorithms
Signal, noise
Source separation
Statistics
Studies
Symmetric matrices
Telecommunications and information theory
Tensile stress
Transformations
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Summary:Simultaneous diagonalization of a set of matrices is a technique that has numerous applications in statistical signal processing and multivariate statistics. Although objective functions in a least-squares sense can be easily formulated, their minimization is not trivial, because constraints and fourth-order terms are usually involved. Most known optimization algorithms are, therefore, subject to certain restrictions on the class of problems: orthogonal transformations, sets of symmetric, Hermitian or positive definite matrices, to name a few. In this paper, we present a new algorithm called QDIAG that splits the overall optimization problem into a sequence of simpler second order subproblems. There are no restrictions imposed on the transformation matrix, which may be nonorthogonal, indefinite, or even rectangular, and there are no restrictions regarding the symmetry and definiteness of the matrices to be diagonalized, except for one of them. We apply the new method to second-order blind source separation and show that the algorithm converges fast and reliably. It allows for an implementation with a complexity independent of the number of matrices and, therefore, is particularly suitable for problems dealing with large sets of matrices
ISSN:1053-587X
1941-0476
DOI:10.1109/TSP.2006.877673