Steepest Descent Algorithms for Optimization Under Unitary Matrix Constraint

• Published in: IEEE transactions on signal processing Vol. 56; no. 3; pp. 1134 - 1147
• Main Authors: Abrudan, T.E., Eriksson, J., Koivunen, V.
• Format: Journal Article
• Language: English
• Published: New York, NY IEEE 01.03.2008; Institute of Electrical and Electronics Engineers; The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
• Subjects: Algorithms; Applied sciences; Array processing; Array signal processing; Complexity; Constraint optimization; Cost function; Descent; Detection, estimation, filtering, equalization, prediction; Exact sciences and technology; Geometry; Independent component analysis; Information, signal and communications theory; Lie groups; Mathematical analysis; Matrices; Matrix methods; MIMO; Miscellaneous; Optimization; Optimization methods; Signal and communications theory; Signal processing; Signal processing algorithms; Signal, noise; source separation; Space technology; Studies; Subspace constraints; subspace estimation; Telecommunications and information theory; unitary matrix constraint; Performance evaluation; Diagonalization; Parameter estimation; Source separation; Array signal processing; Array processing; Steepest descent method; Signal estimation; Blind source separation; optimization; Cost function; Numerical stability; MIMO system; Unitary matrix; unitary matrix constraint; Computer simulation; Geodesy; Theoretical study; Algorithm; Computational complexity; Complex variable method; Constrained optimization; Real function; Geodesic; Signal processing; Lie group; subspace estimation
• ISSN: 1053-587X; 1941-0476
• DOI: 10.1109/TSP.2007.908999

In many engineering applications we deal with constrained optimization problems with respect to complex-valued matrices. This paper proposes a Riemannian geometry approach for optimization of a real-valued cost function T of complex-valued matrix argument W, under the constraint that W is an n times n unitary matrix. We derive steepest descent (SD) algorithms on the Lie group of unitary matrices U(n). The proposed algorithms move towards the optimum along the geodesics, but other alternatives are also considered. We also address the computational complexity and the numerical stability issues considering both the geodesic and the nongeodesic SD algorithms. Armijo step size [1] adaptation rule is used similarly to [2], but with reduced complexity. The theoretical results are validated by computer simulations. The proposed algorithms are applied to blind source separation in MIMO systems by using the joint diagonalization approach [3]. We show that the proposed algorithms outperform other widely used algorithms.