Riemannian-geometric optimization methods for MIMO multiple access channels

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.