Combinatorial Message Sharing and a New Achievable Region for Multiple Descriptions

Abstract

This paper presents a new achievable rate-distortion region for the general $L$ channel multiple descriptions (MDs) problem. A well-known general region for this problem is due to Venkataramani, Kramer, and Goyal (VKG). Their encoding scheme is an extension of the El Gamal-Cover (EC) and Zhang–Berger (ZB) coding schemes to the $L$ channel case and includes a combinatorial number of refinement codebooks, one for each subset of the descriptions. As in ZB, the scheme also allows for a single common codeword to be shared by all descriptions. This paper proposes a novel encoding technique involving combinatorial message sharing (CMS), where every subset of the descriptions may share a distinct common message. This introduces a combinatorial number of shared codebooks along with the refinement codebooks of. These shared codebooks provide a more flexible framework to tradeoff redundancy across the messages for resilience to descriptions loss. We derive an achievable rate-distortion region for the proposed technique, and show that it subsumes the VKG region for general sources and distortion measures. We further show that CMS provides a strict improvement of the achievable region for any source and distortion measures for which some two-description subset is such that ZB achieves points outside the EC region. We then show a more surprising result: CMS outperforms VKG for a general class of sources and distortion measures, including scenarios where the ZB and EC regions coincide for all two-description subsets. In particular, we show that CMS strictly improves on VKG, for the $L$ -channel quadratic Gaussian MD problem, for all $L\geq 3$ , despite the fact that the EC region is complete for the corresponding two-descriptions problem. Consequently, the correlated quantization scheme (an extreme special case of VKG) that has been proven to be optimal for several cross sections of the $L$ -channel quadratic Gaussian MD problem is strictly suboptimal in general. Using the encoding principles derived, we show that the CMS scheme achieves the complete rate-distortion region for several asymmetric cross sections of the $L$ -channel quadratic Gaussian MD problem.