Reconstructing Gaussian Sources by Spatial Sampling

Consider a Gaussian memoryless multiple source (GMMS) with m components with joint probability distribution known only to lie in a given class of distributions. A subset of k ≤ m components is 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.