On Distributed Compression of Linear Functions

Distributed compression of a pair of Gaussian sources in which the goal is to reproduce a linear function of the sources at the decoder is considered. It has recently been noted that lattice codes can provide improved compression rates for this problem compared to conventional, unstructured codes. It is first shown that the state-of-the-art lattice scheme can be improved by including an additional linear binning stage. An outer bound on the rate-distortion region and a separate lower bound on the optimal sum rate are then established. The outer bound implies that for the special case of communicating the difference of two positively correlated Gaussian sources, the unimproved lattice scheme achieves within one bit of the rate region at any distortion level. The sum rate lower bound implies that unstructured codes achieve within one bit of the optimal sum rate whenever the weights of the two sources in the linear combination differ by more than a factor of two.