Riemannian-geometric optimization methods for MIMO multiple access channels

Panayotis Mertikopoulos, Aris L. Moustakas

Abstract: Drawing ideas from Riemannian geometry, we develop a distributed optimization dynamical system for determining optimum input signal covariance matrices in MIMO multiple access channels. In this type of problems, standard (Euclidean) gradient ascent approaches fail because the problem's semidefiniteness constraints are generically violated along the gradient flow; however, by endowing the space of positive-definite matrices with a non-Euclidean geometry which becomes singular when the eigenvalues of the users' covariance matrices approach zero, we are able to derive a matrix-valued Riemannian gradient ascent scheme which converges to the system's optimum transmit spectrum. More to the point, we show that by tuning the geometry of the semidefinite cone, the algorithm's convergence speed changes significantly. As a result, for a specific choice of geometry (which extends the well-known replicator dynamics of evolutionary game theory to a matrix setting), our scheme converges within a few iterations and users are able to track the optimum signal profile even in the presence of rapidly changing channel conditions.