The Gaussian Many-Help-One Distributed Source Coding Problem

Jointly Gaussian memoryless sources (y1,...,yN) are observed at N 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 y1, 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 ditributed quantization rate tuples and the distortion in reconstruction: each encoder consists of a point-to-point vector quantizer followed by a Slepian-Wolf binning encoder.