The Gaussian many-help-one distributed source coding problem

Jointly Gaussian memoryless sources are observed at Ν distinct terminals. The goal is to efficiently encode the observations in a distributed fashion so as to enable reconstruction of any one of the observations, say the first one, at the decoder subject to a quadratic fidelity criterion. Our main result is a precise characterization of the rate-distortion region when the covariance matrix of the sources satisfies a "tree-structure" condition. In this situation, a natural analog-digital separation scheme optimally trades off the distributed quantization rate tuples and the distortion in the reconstruction: Each encoder consists of a point-to-point Gaussian vector quantizer followed by a Slepian-Wolf binning encoder. We also provide a partial converse that suggests that the tree-structure condition is fundamental.