  Consider a lossy compression system with $\ell$ distributed encoders and a
centralized decoder. Each encoder compresses its observed source and forwards
the compressed data to the decoder for joint reconstruction of the target
signals under the mean squared error distortion constraint. It is assumed that
the observed sources can be expressed as the sum of the target signals and the
corruptive noises, which are generated independently from two symmetric
multivariate Gaussian distributions. Depending on the parameters of such
distributions, the rate-distortion limit of this system is characterized either
completely or at least for sufficiently low distortions. The results are
further extended to the robust distributed compression setting, where the
outputs of a subset of encoders may also be used to produce a non-trivial
reconstruction of the corresponding target signals. In particular, we obtain in
the high-resolution regime a precise characterization of the minimum achievable
reconstruction distortion based on the outputs of $k+1$ or more encoders when
every $k$ out of all $\ell$ encoders are operated collectively in the same mode
that is greedy in the sense of minimizing the distortion incurred by the
reconstruction of the corresponding $k$ target signals with respect to the
average rate of these $k$ encoders.
