Distributed Compression of Linear Functions: Partial Sum-Rate Tightness and Gap to Optimal Sum-Rate

We consider the problem of distributed compression of the difference Z = Y1 - cY2 of two jointly Gaussian sources Y1 and Y2 under an MSE distortion constraint D on Z. The rate region for this problem is unknown if the correlation coefficient ρ and the weighting factor c satisfy cρ > 0. Inspired by Ahlswede and Han's scheme for the problem of distributed compression of the modulo-2 sum of two binary sources, we first propose a hybrid random-structured coding scheme that is capable of saving the sum-rate over both the random quantize-and-bin (QB) coding scheme and Krithivasan and Pradhan's structured lattice coding scheme. The main idea is to use a random coding component in the first layer to adjust the source correlation so that the structured coding component in the second layer can be more efficient with the outputs from the first layer as decoder side information. We then provide a new sum-rate lower bound for the problem in hand by connecting it to the Gaussian two-terminal source coding problem with covariance matrix distortion constraint. Our lower bound not only improves existing bounds in many cases, but also allows us to prove sum-rate tightness of the QB scheme when c is either relatively small or large and D is larger than some threshold. Furthermore, our lower bound enables us to show that our new hybrid scheme performs within two b/s from the optimal sum-rate for all values of ρ, c, and D.