Sum-set Inequalities from Aligned Image Sets: Instruments for Robust GDoF Bounds

We present sum-set inequalities specialized to the generalized degrees of freedom (GDoF) framework. These are information theoretic lower bounds on the entropy of bounded density linear combinations of discrete, power-limited dependent random variables in terms of the joint entropies of arbitrary linear combinations of new random variables that are obtained by power level partitioning of the original random variables. These bounds generalize the aligned image sets approach, and are useful instruments to obtain GDoF characterizations for wireless networks, especially with multiple antenna nodes, subject to arbitrary channel strength and channel uncertainty levels. To demonstrate the utility of these bounds, we consider a non-trivial instance of wireless networks - a two user interference channel with different number of antennas at each node, and different levels of partial channel knowledge available to the transmitters. We obtain tight GDoF characterization for specific instance of this channel with the aid of sum-set inequalities.