Combinatorial Message Sharing and a New Achievable Region for Multiple Descriptions

TL;DR

This paper introduces a novel combinatorial message sharing technique for multiple descriptions coding, expanding the achievable rate-distortion region beyond existing schemes, and demonstrates its superiority for Gaussian sources with multiple descriptions.

Contribution

It proposes a new encoding method called Combinatorial Message Sharing (CMS) that generalizes previous schemes and improves achievable regions for multiple descriptions problems.

Findings

CMS subsumes the VKG region for general sources.

CMS strictly outperforms VKG for certain sources and distortions.

CMS achieves the complete rate-distortion region for specific Gaussian cases.

Abstract

This paper presents a new achievable rate-distortion region for the general L channel multiple descriptions problem. A well known general region for this problem is due to Venkataramani, Kramer and Goyal (VKG) [1]. 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 [1]. 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 2-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 2-description subsets. In particular, we show that CMS strictly improves on VKG, for the L-channel quadratic Gaussian MD problem, for all L greater than or equal to 3, despite the fact that the EC region is complete for the corresponding 2-descriptions problem. 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.