The ADMM Penalized Decoder for LDPC Codes

Abstract

Linear programming (LP) decoding for low-density parity-check (LDPC) codes proposed by Feldman et al. is shown to have theoretical guarantees in several regimes and empirically is not observed to suffer from an error floor. However at low signal-to-noise ratios (SNRs), LP decoding is observed to have worse error performance than belief propagation (BP) decoding. In this paper, we seek to improve LP decoding at low SNRs while still achieving good high SNR performance. We first present a new decoding framework obtained by trying to solve a nonconvex optimization problem using the alternating direction method of multipliers (ADMM). This non-convex problem is constructed by adding a penalty term to the LP decoding objective. The goal of the penalty term is to make "pseudocodewords", which are the non-integer vertices of the LP relaxation to which the LP decoder fails, more costly. We name this decoder class the "ADMM penalized decoder". In our simulation results, the ADMM penalized decoder with ℓ 1 and ℓ 2 penalties outperforms both BP and LP decoding at all SNRs. For high SNR regimes where it is infeasible to simulate, we use an instanton analysis and show that the ADMM penalized decoder has better high SNR performance than BP decoding. We also develop a reweighted LP decoder using linear approximations to the objective with an ℓ 1 penalty. We show that this decoder has an improved theoretical recovery threshold compared to LP decoding. In addition, we show that the empirical gain of the reweighted LP decoder is significant at low SNRs.

Improved ADMM Penalized Decoder for Irregular Low-Density Parity-Check Codes

“…Before we go through the parameter optimization for the proposed decoder, we make a remark for the all-zeros assumption of the improved ADMM penalized decoder. Following the proof of the all-zeros assumption for the ADMM penalized decoder [7], we can see that the improved ADMM penalized decoder also satisfy Lemma 18 in [7]. Then similar arguments can be used to show that Lemma 19 and Lemma 16 in [7] also hold for the improved ADMM penalized decoder.…”

“…Following the proof of the all-zeros assumption for the ADMM penalized decoder [7], we can see that the improved ADMM penalized decoder also satisfy Lemma 18 in [7]. Then similar arguments can be used to show that Lemma 19 and Lemma 16 in [7] also hold for the improved ADMM penalized decoder. Therefore, the improved ADMM penalized decoder satisfies the all-zeros assumption.…”

“…and (2) How to choose penalty coefficients in high SNR regimes? Based on instanton analysis, Liu et al investigate the relationship between the instanton norm and the penalty coefficient for the ADMM penalized decoder [7]. It provides a method to choose penalty coefficient in high SNR regimes.…”

“…called ADMM penalized decoder [5]- [7]. The goal of the penalty term is to make pseudocodewords more costly.…”

Multicore implementation of LDPC decoders based on ADMM algorithm

“…Besides, it is observed that the ADMM-based LP decoder has worse error performance than the MP decoder at low SNR values [25]. To address this problem, Liu et al proposed a penalized ADMM decoder which adds a penalty term into the objective function of the LP formulation.…”