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Steepest Descent Algorithms for Optimization Under Unitary Matrix Constraint

Domenii publicaţii > Stiinte ingineresti + Tipuri publicaţii > Articol în revistã ştiinţificã

Autori: T. Abrudan and J. Eriksson and V. Koivunen

Editorial: IEEE, IEEE Transactions on Signal Processing, 56(3), p.1134-1147, 2008.


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 {mathcal J} of complex-valued matrix argument {mathbf W}, under the constraint that {mathbf 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.

Cuvinte cheie: - // Array processing, optimization, source separation, subspace estimation, unitary matrix constraint.