Geometric Mean Decomposition and Generalized Triangular Decomposition

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

H = QRP* • Matrix Decomposition H = QRP* • Q, P: matrices with orthonormal columns • R: upper triangular • 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 SIAM 2004

Motivation–Joint Transceiver Design for MIMO Communications • Multi-input Multi-output Communications Received data Channel matrix SIAM 2004

MIMO Transceiver • MIMO Transceiver • Decomposition H=QRP* • Linear precoder P , x = P s • Linear equalizer Q, v = Q*y • Equivalent Channel • v = R s + Q* z • Overall System Performance Limited by SIAM 2004

Problem Formulation • Generalized Maximin problem • The Solution is Geometric Mean Decomposition • P, Q: matrices with orthonormal columns • R: upper triangular with equal diagonal elements SIAM 2004

GMD Algorithm • Starts with SVD • Applies Givens Rotations and Permutations to • If • Illustration of k-th step • K-1 iterations • Non-unique SIAM 2004

A Numerical Research Problem • 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] • GMD is a new solution to Higham’s problem SIAM 2004

Advantages • Advantages of GMD • 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 • Numerically stable – involves 2K-2 Givens rotations • The technique can be easily extended to the generalized triangular decomposition (GTD) SIAM 2004

GTD : H = QRP* • 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 O1: H and R share the same singular values O2: The diagonal are the eigenvalues of R SIAM 2004

Weyl-Horn Theorem • Weyl-Horn Theorem SIAM 2004

GTD Theorem • Generalized Triangular Decomposition [Jiang et. al. 2004] SIAM 2004

GTD Algorithm • Starts with SVD • Applies Givens Rotations and Permutations to • If • Illustration of k-th step • K-1 iterations • Key difference from GMD is the permutation matrices SIAM 2004

Applications of GTD • 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] SIAM 2004

Geometric Mean Decomposition and Generalized Triangular Decomposition