Strange bedfellows: Riemann, Gibbs and vector Gaussian multiple access channels

Panayotis Mertikopoulos 1
1 MESCAL - Middleware efficiently scalable
Inria Grenoble - Rhône-Alpes, LIG - Laboratoire d'Informatique de Grenoble
Abstract : 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 semidefiniteness constraints, we endow the space of positive-definitie 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).
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https://hal.inria.fr/hal-00788010
Contributor : Arnaud Legrand <>
Submitted on : Wednesday, February 13, 2013 - 3:02:35 PM
Last modification on : Thursday, November 8, 2018 - 2:28:02 PM

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Panayotis Mertikopoulos. Strange bedfellows: Riemann, Gibbs and vector Gaussian multiple access channels. NetGCoop'12: Proceedings of the 6th International Conference on Network Games, Control and Optimization, 2012, Unknown. ⟨hal-00788010⟩

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