Iterative Quantization Using Codes On Graphs

TL;DR

This paper explores the use of graph-based codes and iterative message passing algorithms for binary erasure quantization, demonstrating that duals of capacity-achieving channel codes can approach optimal source coding rates.

Contribution

It shows that duals of capacity-achieving codes for the BEC can be used for near-optimal binary erasure quantization, highlighting the role of duality in code design.

Findings

Duals of capacity-achieving BEC codes approach minimal BEQ rate.

LDPC codes cannot achieve minimal BEQ rate unless density grows logarithmically.

Duals of BEC decoding algorithms enable efficient BEQ encoding.

Abstract

We study codes on graphs combined with an iterative message passing algorithm for quantization. Specifically, we consider the binary erasure quantization (BEQ) problem which is the dual of the binary erasure channel (BEC) coding problem. We show that duals of capacity achieving codes for the BEC yield codes which approach the minimum possible rate for the BEQ. In contrast, low density parity check codes cannot achieve the minimum rate unless their density grows at least logarithmically with block length. Furthermore, we show that duals of efficient iterative decoding algorithms for the BEC yield efficient encoding algorithms for the BEQ. Hence our results suggest that graphical models may yield near optimal codes in source coding as well as in channel coding and that duality plays a key role in such constructions.