The Quadratic Gaussian Rate-Distortion Function for Source Uncorrelated Distortions

TL;DR

This paper derives the rate-distortion function for Gaussian sources with the constraint that the distortion remains uncorrelated to the source, revealing a larger rate than the classical function and characterizing the optimal distortion statistics.

Contribution

It provides a complete characterization of the rate-distortion function under the uncorrelated distortion constraint for Gaussian sources, including the structure of the optimal solution.

Findings

The rate-distortion function with uncorrelated distortion is strictly larger than the classical one.

The solution involves coupled equations similar to water-filling.

The gap between constrained and unconstrained rates increases with distortion.

Abstract

We characterize the rate-distortion function for zero-mean stationary Gaussian sources under the MSE fidelity criterion and subject to the additional constraint that the distortion is uncorrelated to the input. The solution is given by two equations coupled through a single scalar parameter. This has a structure similar to the well known water-filling solution obtained without the uncorrelated distortion restriction. Our results fully characterize the unique statistics of the optimal distortion. We also show that, for all positive distortions, the minimum achievable rate subject to the uncorrelation constraint is strictly larger than that given by the un-constrained rate-distortion function. This gap increases with the distortion and tends to infinity and zero, respectively, as the distortion tends to zero and infinity.