On the Gaussian K-description problem under symmetric distortion constraints

Abstract

We consider multiple description (MD) coding for the Gaussian source under the symmetric mean squared error distortion constraints. With focus on the three description problem, we provide inner and outer bounds for the rate region, between which the gap can be bounded by some small constants. At the heart of this result is a novel lower bound for the sum rate, which is derived through generalization of the well-known bounding technique by Ozarow. In contrast to the original method, we expand the probability space by more than one (instead of only one) random variable, and further impose a particular Markov structure on them. The outer bound is then established by applying this technique to several bounding planes of the rate region. For the inner bound, we consider a simple scheme of combining successive refinement coding and lossless multilevel diversity coding (MLD). Both the inner and outer bounds can be written as the intersection of ten half spaces with matching normal directions, and thus can be easily compared. The small gap between them, where the boundary of the MD rate region clearly resides, suggests the surprising competitiveness of this simple achievability scheme. The geometric structure of the MLD rate region provides important guidelines as to the normal directions of the outer bound hyperplanes, which demonstrates an intimate connection between MD and MLD coding. These results can be generalized and improved in various ways which are also discussed.