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Register a new userLearned Decimation for Neural Belief Propagation Decoders
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Added by
Andreas Buchberger
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We introduce a two-stage decimation process to improve the performance of neural belief propagation (NBP), recently introduced by Nachmani et al., for short low-density parity-check (LDPC) codes. In the first stage, we build a list by iterating between a conventional NBP decoder and guessing the least reliable bit. The second stage iterates between a conventional NBP decoder and learned decimation, where we use a neural network to decide the decimation value for each bit. For a (128,64) LDPC code, the proposed NBP with decimation outperforms NBP decoding by 0.75 dB and performs within 1 dB from maximum-likelihood decoding at a block error rate of $10^{-4}$.
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We consider near maximum-likelihood (ML) decoding of short linear block codes. In particular, we propose a novel decoding approach based on neural belief propagation (NBP) decoding recently introduced by Nachmani et al. in which we allow a different parity-check matrix in each iteration of the algorithm. The key idea is to consider NBP decoding over an overcomplete parity-check matrix and use the weights of NBP as a measure of the importance of the check nodes (CNs) to decoding. The unimportant CNs are then pruned. In contrast to NBP, which performs decoding on a given fixed parity-check matrix, the proposed pruning-based neural belief propagation (PB-NBP) typically results in a different parity-check matrix in each iteration. For a given complexity in terms of CN evaluations, we show that PB-NBP yields significant performance improvements with respect to NBP. We apply the proposed decoder to the decoding of a Reed-Muller code, a short low-density parity-check (LDPC) code, and a polar code. PB-NBP outperforms NBP decoding over an overcomplete parity-check matrix by 0.27-0.31 dB while reducing the number of required CN evaluations by up to 97%. For the LDPC code, PB-NBP outperforms conventional belief propagation with the same number of CN evaluations by 0.52 dB. We further extend the pruning concept to offset min-sum decoding and introduce a pruning-based neural offset min-sum (PB-NOMS) decoder, for which we jointly optimize the offsets and the quantization of the messages and offsets. We demonstrate performance 0.5 dB from ML decoding with 5-bit quantization for the Reed-Muller code.
We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate-distortion limit. The Belief Propagation Guided Decimation encoder is based on the posterior measure of a binary symmetric test-channel. This measure can be interpreted as a random Gibbs measure at a temperature directly related to the noise level of the test-channel. We investigate the links between the algorithmic performance of the Belief Propagation Guided Decimation encoder and the phase diagram of this Gibbs measure. The phase diagram is investigated thanks to the cavity method of spin glass theory which predicts a number of phase transition thresholds. In particular the dynamical and condensation phase transition temperatures (equivalently test-channel noise thresholds) are computed. We observe that: (i) the dynamical temperature of the spatially coupled construction saturates towards the condensation temperature; (ii) for large degrees the condensation temperature approaches the temperature (i.e. noise level) related to the information theoretic Shannon test-channel noise parameter of rate-distortion theory. This provides heuristic insight into the excellent performance of the Belief Propagation Guided Decimation algorithm. The paper contains an introduction to the cavity method.
This paper focuses on finite-dimensional upper and lower bounds on decodable thresholds of Zm and binary low-density parity-check (LDPC) codes, assuming belief propagation decoding on memoryless channels. A concrete framework is presented, admitting systematic searches for new bounds. Two noise measures are considered: the Bhattacharyya noise parameter and the soft bit value for a maximum a posteriori probability (MAP) decoder on the uncoded channel. For Zm LDPC codes, an iterative m-dimensional bound is derived for m-ary-input/symmetric-output channels, which gives a sufficient stability condition for Zm LDPC codes and is complemented by a matched necessary stability condition introduced herein. Applications to coded modulation and to codes with non-equiprobable distributed codewords are also discussed. For binary codes, two new lower bounds are provided for symmetric channels, including a two-dimensional iterative bound and a one-dimensional non-iterative bound, the latter of which is the best known bound that is tight for binary symmetric channels (BSCs), and is a strict improvement over the bound derived by the channel degradation argument. By adopting the reverse channel perspective, upper and lower bounds on the decodable Bhattacharyya noise parameter are derived for non-symmetric channels, which coincides with the existing bound for symmetric channels.
Belief-propagation (BP) decoders play a vital role in modern coding theory, but they are not suitable to decode quantum error-correcting codes because of a unique quantum feature called error degeneracy. Inspired by an exact mapping between BP and deep neural networks, we train neural BP decoders for quantum low-density parity-check (LDPC) codes with a loss function tailored to error degeneracy. Training substantially improves the performance of BP decoders for all families of codes we tested and may solve the degeneracy problem which plagues the decoding of quantum LDPC codes.
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding threshold is as large as possible. We introduce a new code optimization technique which involves the search space range which can be thought of as minimizing randomness in differential evolution or limiting the search range in exhaustive search. This technique is applied to the design of good irregular LDPC codes and multiedge type LDPC codes.
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