On the Computation of the Capacity Region of the Discrete MAC

Abstract

The computation of the channel capacity of discrete memoryless channels is a convex problem that can be efficiently solved using the Arimoto-Blahut (AB) iterative algorithm. However, the extension of this algorithm to the computation of capacity regions of multiterminal networks is not straightforward since it gives rise to non-convex problems. In this context, the AB algorithm has only been successfully extended to the calculation of the sum-capacity of the discrete memoryless multiple-access channel (DMAC). Thus, the computation of the whole capacity region still requires the use of computationally demanding search methods. In this paper, we first give an alternative reformulation of the capacity region of the DMAC which condenses all the non-convexities of the problem into a single rank-one constraint. Then, we propose efficient methods to compute outer and inner bounds on the capacity region of the two-user DMAC by solving a relaxed version of the problem and projecting its solution onto the original feasible set. Targeting numerical results, we first take a randomization approach. Focusing on analytical results, we study projection via minimum divergence, which amounts to the marginalization of the relaxed solution. In this case we derive sufficient conditions and necessary and sufficient conditions for the bounds to be tight. Furthermore, we are able to show that the class of channels for which the marginalization bounds match exactly the capacity region includes all the two-user binary-input deterministic DMACs as well as other non-deterministic channels. In general, however, both methods are able to compute very tight bounds as shown for various examples.