Reconstructing Gaussian sources by spatial sampling

TL;DR

This paper investigates the optimal tradeoffs in sampling and compressing Gaussian sources to enable accurate reconstruction of all components, introducing a universal rate distortion function and revealing a structural property for efficient encoding.

Contribution

It introduces the universal sampling rate distortion function for Gaussian sources and characterizes it with single-letter formulas, highlighting the optimality of sequential sampling and reconstruction.

Findings

Single-letter characterizations of the universal sampling rate distortion function.

Optimality of compressing sampled components first, then estimating unsampled ones.

Structural property guiding efficient sampling and compression strategies.

Abstract

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.