P. Mertikopoulos. In NetGCoop '12: Proceedings of the 6th International Conference on Network Games, Control and Optimization, 2012.
Using tools and techniques from Riemannian geometry, we develop a novel distributed algorithm for optimizing the input signal distribution (and, in particular, its covariance matrix) in Gaussian multiple-input, multiple-output multiple access channels. To account for the problem’s semi-definiteness constraints, we endow the space of positive-definite matrices with a non-Euclidean spectral metric which becomes singular when the signal spectrum itself becomes singular. Quite remarkably, viewing the unit simplex as a subspace of the space of semidefinite matrices (corresponding to diagonal ones), we show that this metric generalizes the well-known Shahshahani metric on the simplex and extends the replicator dynamics of evolutionary game theory; in fact, gradient ascent trajectories defined with respect to this metric are shown to be equivalent to a Gibbs-based exponential learning process. In this way, we show that the resulting optimization algorithm converges to the optimum signal distribution exponentially fast: users attain an ε-neighborhood of the system’s optimum configuration in time which is at most O(log(1/ε)) (and, in practice, within only a few iterations, even for large numbers of users).