Probabilistic Analysis of Linear Programming Decoding

TL;DR

This paper provides a probabilistic analysis demonstrating that LP decoding of LDPC codes can correct a constant fraction of errors with high probability, surpassing previous finite-length results and introducing new combinatorial tools.

Contribution

It introduces a novel probabilistic analysis framework for LP decoding, including a combinatorial characterization and the concept of probabilistic expansion in bipartite graphs.

Findings

LP decoding corrects a constant fraction of errors with high probability.

Analysis exceeds previous non-asymptotic and finite-length results by over ten times.

Introduces probabilistic expansion in bipartite graphs as a key concept.

Abstract

We initiate the probabilistic analysis of linear programming (LP) decoding of low-density parity-check (LDPC) codes. Specifically, we show that for a random LDPC code ensemble, the linear programming decoder of Feldman et al. succeeds in correcting a constant fraction of errors with high probability. The fraction of correctable errors guaranteed by our analysis surpasses previous non-asymptotic results for LDPC codes, and in particular exceeds the best previous finite-length result on LP decoding by a factor greater than ten. This improvement stems in part from our analysis of probabilistic bit-flipping channels, as opposed to adversarial channels. At the core of our analysis is a novel combinatorial characterization of LP decoding success, based on the notion of a generalized matching. An interesting by-product of our analysis is to establish the existence of ``probabilistic expansion'' in random bipartite graphs, in which one requires only that almost every (as opposed to every) set of a certain size expands, for sets much larger than in the classical worst-case setting.