Riemannian Geometry Based Methods for Optimization under Unitary Matrix Constraint

Visa Koivunen, Ph.D.

Visiting Fellow
Princeton University (on sabbatical leave from Helsinki Univ. of Technology, Finland)

Abstract:

In many engineering applications we deal with constrained optimization problems with respect to complex valued matrices. We propose a Riemannian geometry approach for optimization of a real valued cost function J of a complex valued matrix W, under the constraint that W is an n-by-n unitary matrix. Such optimization has applications in beamforming, space-time coding, subspace estimation, signal separation as well as MIMO systems.

We derive steepest descent and conjugate gradient algorithms on the Lie group of unitary matrices U(n). The algorithms satisfy the constraint exactly unlike methods based on Lagrange multipliers. The proposed algorithms move towards the optimum along the geodesics, but other alternatives are also considered. We also address the computational complexity considering both the geodesic and the non-geodesic steepest descent algorithms. Armijo step size adaptation rule is used. Simulation examples in MIMO communication systems are provided.

Biography:

Visa Koivunen (Senior Member, IEEE) received his D.Sc. (Tech) degree with honors from the University of Oulu, Dept. of Electrical Engineering. He received the primus doctor (best graduate) award among the doctoral graduates in years 1989-1994. From 1992 to 1995, he was a visiting researcher at the University of Pennsylvania, Philadelphia, USA. In 1996, he held a faculty position at the Department of Electrical Engineering at the University of Oulu. From August 1997 to August 1999, he was an Associate Professor at the Signal Processing Labroratory at Tampere University of Technology. Since 1999, he has been a Professor of Signal Processing at the Department of Electrical and Communications Engineering at Helsinki University of Technology (HUT) in Finland. He is one of the Principal Investigators in SMARAD (Smart and Novel Radios) Center of Excellence in Radio and Communications Engineering nominated by the Academy of Finland. Since 2003, he has also been an adjunct full professor at the University of Pennsylvania in Philadelphia, PA. During his sabbatical leave in 2006-2007, he was a Nokia Visiting Fellow at the Nokia Research Center as well as a Visiting Fellow at Princeton University.

Dr. Koivunen's research interests include statistical, communications and sensor array signal processing. He has published more than 200 papers in international scientific conferences and journals. He co-authored the papers receiving the best paper award in IEEE PIMRC 2005, EUSIPCO 2006 and EuCAP 2006. He served as an associate editor for IEEE Signal Processing Letters. He is a member of the editorial board for the Signal Processing Journal and the Journal of Wireless Communication and Networking. He is also a member of the IEEE Signal Processing for Communication Technical Committee (SPCOM-TC). He is the general chair of the IEEE SPAWC (Signal Processing Advances in Wireless Communication) 2007 conference in Helsinki in June 2007.

Friday, May 25th at 12 p.m.