An outer bound for distributed compression of linear functions

We consider the problem of distributed compression of a pair of Gaussian sources in which the goal is to reproduce a given linear combination of the pair, subject to an upper bound on the allowable mean-square error. The rate-distortion region for this problem is unknown except in certain cases, and several achievable schemes have been proposed. We provide an outer bound to the rate-distortion region, which strictly improves upon the cut-set bound. For the special case of reproducing the difference of two positively correlated Gaussian random variables, the outer bound determines the rate region to within one bit at all distortion levels.