Approximating the Gaussian Multiple Description Rate Region Under Symmetric Distortion Constraints

TL;DR

This paper provides an approximate characterization of the Gaussian multiple description rate region under symmetric distortion constraints, establishing bounds and a constant gap, and extends results to general sources.

Contribution

It introduces a polytopic approximation of the rate region and generalizes Ozarow's technique to bound the gap for multiple descriptions.

Findings

The rate region can be bounded between two polytopes with a constant gap.

The maximum gap for individual description rate is no larger than 0.92 bits.

The separation approach combining successive refinement and lossless coding is near-optimal.

Abstract

We consider multiple description coding for the Gaussian source with K descriptions under the symmetric mean squared error distortion constraints, and provide an approximate characterization of the rate region. We show that the rate region can be sandwiched between two polytopes, between which the gap can be upper bounded by constants dependent on the number of descriptions, but independent of the exact distortion constraints. Underlying this result is an exact characterization of the lossless multi-level diversity source coding problem: a lossless counterpart of the MD problem. This connection provides a polytopic template for the inner and outer bounds to the rate region. In order to establish the outer bound, we generalize Ozarow's technique to introduce a strategic expansion of the original probability space by more than one random variables. For the symmetric rate case with any number of descriptions, we show that the gap between the upper bound and the lower bound for the individual description rate is no larger than 0.92 bit. The results developed in this work also suggest the "separation" approach of combining successive refinement quantization and lossless multi-level diversity coding is a competitive one, since it is only a constant away from the optimum. The results are further extended to general sources under the mean squared error distortion measure, where a similar but looser bound on the gap holds.