Analytical Framework of LDGM-based Iterative Quantization with Decimation

TL;DR

This paper provides a rigorous theoretical analysis of LDGM-based iterative quantizers with decimation, demonstrating conditions under which belief propagation marginals approximate true marginals well, supporting near-ideal distortion performance.

Contribution

It proves that under certain degree distribution conditions, BP marginals in LDGM quantizers converge to true marginals as block length and iterations grow, validating prior optimization methods.

Findings

BP marginals have vanishing mean-square error asymptotically

Degree distribution conditions can be evaluated with density evolution

Supports near-ideal distortion performance in large-scale settings

Abstract

While iterative quantizers based on low-density generator-matrix (LDGM) codes have been shown to be able to achieve near-ideal distortion performance with comparatively moderate block length and computational complexity requirements, their analysis remains difficult due to the presence of decimation steps. In this paper, considering the use of LDGM-based quantizers in a class of symmetric source coding problems, with the alphabet being either binary or non-binary, it is proved rigorously that, as long as the degree distribution satisfies certain conditions that can be evaluated with density evolution (DE), the belief propagation (BP) marginals used in the decimation step have vanishing mean-square error compared to the exact marginals when the block length and iteration count goes to infinity, which potentially allows near-ideal distortion performances to be achieved. This provides a sound theoretical basis for the degree distribution optimization methods previously proposed in the literature and already found to be effective in practice.