Jointly Gaussian memoryless sources are observed at N distinct terminals. Thegoal is to efficiently encode the observations in a distributed fashion so asto enable reconstruction of any one of the observations, say the first one, atthe decoder subject to a quadratic fidelity criterion. Our main result is aprecise characterization of the rate-distortion region when the covariancematrix of the sources satisfies a "tree-structure" condition. In thissituation, a natural analog-digital separation scheme optimally trades off thedistributed quantization rate tuples and the distortion in the reconstruction:each encoder consists of a point-to-point Gaussian vector quantizer followed bya Slepian-Wolf binning encoder. We also provide a partial converse thatsuggests that the tree structure condition is fundamental.
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