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 (with positive correlation coefficient ρ and positive c) under an MSE distortion constraint D on Z. The rate region for this problem is unknown. We provide a new lower bound on the minimum sum-rate by utilizing the connection of the above problem with the two-terminal source coding problem with 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 Berger-Tung 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 the improved lattice-based scheme recently introduced in [1] (with the smallest achievable sum-rate) performs within 1.18 b/s from the optimal sum-rate for all values of ρ, c, and D.