1. Geometric Mean Decomposition and Generalized Triangular Decomposition Yi JiangWilliam W. HagerJian Li yjiang@dsp.ufl.eduhager@math.ufl.eduli@dsp.ufl.edu yjiang@dsp.ufl.eduhager@math.ufl.eduli@dsp.ufl.edu Department of Electrical & Computer Engineering Department of Mathematics University of Florida July 14, 2004

2. SIAM 2004 2/13 H = QRP* Matrix Decomposition H = QRP* Matrix Decomposition H = QRP* –Q, P: matrices with orthonormal columns –R: upper triangular Some Special Cases Some Special Cases –Singular value decomposition R: diagonal matrix containing singular values of H –Schur decomposition R: upper triangular with eigenvalues of H on the diagonal –QR decomposition P: identity matrix

3. July 14, 2004SIAM 2004 3/13 Motivation – Joint Transceiver Design for MIMO Communications Multi-input Multi-output Communications Multi-input Multi-output Communications Received data Channel matrix

4. July 14, 2004SIAM 2004 4/13 MIMO Transceiver MIMO Transceiver MIMO Transceiver –Decomposition H=QRP* –Linear precoder P, x = P s –Linear equalizer Q, v = Q*y Equivalent Channel Equivalent Channel –v = R s + Q* z Overall System Performance Limited by Overall System Performance Limited by

5. July 14, 2004SIAM 2004 5/13 Problem Formulation Generalized Maximin problem Generalized Maximin problem The Solution is Geometric Mean Decomposition The Solution is Geometric Mean Decomposition –P, Q: matrices with orthonormal columns –R: upper triangular with equal diagonal elements

6. July 14, 2004SIAM 2004 6/13 GMD Algorithm Starts with SVD Starts with SVD Applies Givens Rotations and Permutations to Applies Givens Rotations and Permutations to –If –Illustration of k-th step –K-1 iterations –Non-unique

7. July 14, 2004SIAM 2004 7/13 A Numerical Research Problem A numerical research problem by Higham (1996) A numerical research problem by Higham (1996) –Develop an efficient algorithm for computing a unit upper triangular K x K matrix with prescribed singular values A solution was given by [Kosowski and Smoktunowicz, Computing, 2000] A solution was given by [Kosowski and Smoktunowicz, Computing, 2000] GMD is a new solution to Higham ’ s problem GMD is a new solution to Higham ’ s problem

8. July 14, 2004SIAM 2004 8/13 Advantages Advantages of GMD Advantages of GMD GMD transceiver has superior performance compared with any other published transceiver schemes GMD transceiver has superior performance compared with any other published transceiver schemes Computationally efficient – needs only an additional O((M+N)K) flops compared with SVD Computationally efficient – needs only an additional O((M+N)K) flops compared with SVD Numerically stable – involves 2K-2 Givens rotations Numerically stable – involves 2K-2 Givens rotations The technique can be easily extended to the generalized triangular decomposition (GTD) The technique can be easily extended to the generalized triangular decomposition (GTD)

9. July 14, 2004SIAM 2004 9/13 GTD : H = QRP* Two Questions Two Questions Q1: What is the achievable set for the diagonal of R ? Q2: Is there a systematic approach to get any achievable decomposition? Two Observations Two Observations O1: H and R share the same singular values O2: The diagonal are the eigenvalues of R

10. July 14, 2004SIAM 2004 10/13 Weyl-Horn Theorem Weyl-Horn Theorem Weyl-Horn Theorem

11. July 14, 2004SIAM 2004 11/13 GTD Theorem Generalized Triangular Decomposition [Jiang et. al. 2004] Generalized Triangular Decomposition [Jiang et. al. 2004]

12. July 14, 2004SIAM 2004 12/13 GTD Algorithm Starts with SVD Starts with SVD Applies Givens Rotations and Permutations to Applies Givens Rotations and Permutations to –If –Illustration of k-th step –K-1 iterations –Key difference from GMD is the permutation matrices

13. July 14, 2004SIAM 2004 13/13 Applications of GTD A New Solution to Inverse Eigenvalue Problem A New Solution to Inverse Eigenvalue Problem –Constructing matrices with prescribed eigenvalues and singular values [Chu, SIAM J. Numer. Anal. 2000] Design MIMO Transceiver with Quality of Service (QoS) Constraints [Jiang, et. al., Asilomar, 2004] Design MIMO Transceiver with Quality of Service (QoS) Constraints [Jiang, et. al., Asilomar, 2004]