Reconstructing Gaussian sources by spatial sampling
This work addresses a theoretical problem in information theory for signal processing and compression, offering incremental advancements in understanding sampling and reconstruction tradeoffs for Gaussian sources.
The paper tackles the problem of reconstructing all components of a Gaussian source from a sampled subset under distortion constraints, introducing a universal sampling rate distortion function and providing single-letter characterizations for optimal tradeoffs among sampling, compression, and distortion.
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.