On distributed compression of linear functions

We consider distributed compression of a pair of Gaussian sources in which the goal is to reproduce a linear function of the sources at the decoder. It has recently been noted that lattice codes can provide improved compression rates for this problem compared to conventional, unstructured codes. We show that by including an additional linear binning stage, the state-of-the-art lattice scheme can be improved, in some cases by an arbitrarily large factor. We then describe a lower bound on the optimal sum rate for the case in which the variance of the linear combination exceeds the variance of one of the sources. This lower bound shows that unstructured codes achieve within one bit of the optimal sum rate at any distortion level. We also describe an outer bound on the rate-distortion region that holds in general, which for the special case of communicating the difference of two positively correlated Gaussian sources shows that the unimproved lattice scheme is within one bit of the rate region at any distortion level.