  Consider a Gaussian memoryless multiple source with $m$ components with joint
probability distribution known only to lie in a given class of distributions. A
subset of $k \leq m$ components are sampled and compressed with the objective
of reconstructing all the $m$ components within a specified level of distortion
under a mean-squared error criterion. In Bayesian and nonBayesian settings, the
notion of universal sampling rate distortion function for Gaussian sources is
introduced to capture the optimal tradeoffs among sampling, compression rate
and distortion level. Single-letter characterizations are provided for the
universal sampling rate distortion function. Our achievability proofs highlight
the following structural property: it is optimal to compress and reconstruct
first the sampled components of the GMMS alone, and then form estimates for the
unsampled components based on the former.
