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IEEE Transactions on Cognitive Communications and Networking

Low-Complexity Neural Belief Propagation Decoding Algorithm Based on Tensor Ring Decomposition

Yuanhui Liang, Chan–Tong Lam, Qingle Wu, Benjamin K. Ng, Sio‐Kei Im

Published Oct 30, 2024• 1 citation
Publisher

Abstract

Neural belief propagation (NBP) decoding can improve the performance of belief propagation (BP) decoding for high-density parity check (HDPC) codes, at the expense of higher memory storage requirement and computational complexity due to the addition of trainable weight coefficients. To reduce the high storage requirement of NBP, the cyclically equivariant neural BP (CENBP) algorithm makes full use of the cyclically invariant property of the cyclic code, optimizes and reuses the weight coefficients of the NBP algorithm, at the expense of further increasing the computational complexity of NBP. In this paper, we propose low-complexity, in terms of both memory storage requirement and computational complexity, NBP and CENBP decoding algorithms based on Tensor Ring (TR) decomposition. First, in order to reduce the memory storage and computational complexity of the NBP algorithm, we propose a TR-based compression algorithm to compress the messages and mathematical calculations in the NBP decoding algorithm, called TR-NBP algorithm. Second, to address the high computational complexity of the CENBP algorithm, we propose to apply TR decomposition-based compression to the odd layers of the CENBP decoding algorithm, called TR-CENBP, to reduce the computational complexity, and further reduce the required memory storage requirement of the CENBP algorithm. Furthermore, we use TR decomposition-based compression to simplify the mathematical computations associated with thetanhfunction in the NBP algorithm to further reduce the complexity of the hardware implementation. Experimental results show that direct compression of BP algorithm using TR decomposition results in significant performance degradation and our proposed low complexity TR-NBP algorithm and TR-CENBP algorithm can greatly reduce both the memory storage requirement and computation complexity, without significant performance degradation for typical BCH and LDPC codes.

DOI
10.1109/tccn.2024.3487999

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Neural belief propagation (NBP) decoding can improve the performance of belief propagation (BP) decoding for high-density parity check (HDPC) codes, at the expense of higher memory storage requirement and computational complexity due to the addition of trainable weight coefficients. To reduce the high storage requirement of NBP, the cyclically equivariant neural BP (CENBP) algorithm makes full use of the cyclically invariant property of the cyclic code, optimizes and reuses the weight coefficients of the NBP algorithm, at the expense of further increasing the computational complexity of NBP. In this paper, we propose low-complexity, in terms of both memory storage requirement and computational complexity, NBP and CENBP decoding algorithms based on Tensor Ring (TR) decomposition. First, in order to reduce the memory storage and computational complexity of the NBP algorithm, we propose a TR-based compression algorithm to compress the messages and mathematical calculations in the NBP decoding algorithm, called TR-NBP algorithm. Second, to address the high computational complexity of the CENBP algorithm, we propose to apply TR decomposition-based compression to the odd layers of the CENBP decoding algorithm, called TR-CENBP, to reduce the computational complexity, and further reduce the required memory storage requirement of the CENBP algorithm. Furthermore, we use TR decomposition-based compression to simplify the mathematical computations associated with thetanhfunction in the NBP algorithm to further reduce the complexity of the hardware implementation. Experimental results show that direct compression of BP algorithm using TR decomposition results in significant performance degradation and our proposed low complexity TR-NBP algorithm and TR-CENBP algorithm can greatly reduce both the memory storage requirement and computation complexity, without significant performance degradation for typical BCH and LDPC codes.
In this paper, we propose low complexity joint-way compression algorithms with Tensor-Ring (TR) decomposition and weight sharing to further lower the storage and computational complexity requirements for low density parity check (LDPC) neural decoding. Compared with Tensor-Train (TT) decomposition, TR decomposition is more flexible for the selection of ranks, and is also conducive to the use of rank optimization algorithms. In particular, we use TR decomposition to decompose not only the weight parameter matrix of Neural Normalized Min-Sum (NNMS)+ algorithm [16], but also the message matrix transmitted between variable nodes and check nodes. Furthermore, we combine the TR decomposition and temporal and spatial weight sharing algorithm, called joint-way compression, to further lower the complexity of LDPC neural decoding algorithm. We show that the joint-way compression algorithm can achieve better compression efficiency than a single compression algorithm, while maintaining a comparable bit error rate (BER) performance. From the numerical experiments, we found that all the compression algorithms with appropriate selection of ranks give almost no performance BER degradation and that the TRwm-ssNNMS+ algorithm, which combines the spatial sharing and TR decomposition of both weight and message matrix, has the best compression performance. Comparing with our TT-NNMS+ algorithm proposed in [16], the number of parameters is reduced by about 70 times and the number of multiplications is reduced by about 6 times.
With the rapid development of deep learning research and physical layer communication applications, deep learning-based channel coding has gradually become a research hotspot. In this paper, we propose a LDPC decoding network based on neural belief propagation (NBP) decoding. By introducing weights sharing mechanism, the weights vary across different iterations. The number of parameters is reduced, which significantly lowers memory requirements. In addition, we optimize the loss function to better train the model, achieving a lower bit error rate (BER) performance. Experimental results show that the proposed decoder yields significant performance improvements with respect to NBP, and significantly reducing the number of learnable weights.
In this letter, we propose a low-complexity hypernetwork for channel neural decoding with learning weights in tensor ring (TR) format, called TRHyper. The internal parameters and the number of layers of the TRHyper based channel neural decoding algorithm can be updated without retraining. We design the size of each TRHyper layer according to the size of the factor tensor in tensor ring format. During the training phase, we reuse the storage space for the learning weights of the main decoding network, so the proposed TRHyper no longer require additional storage space for its learning weights. Numerical results show that for low-density parity check (LDPC) codes, the performance of the TRHyper based channel neural decoder is similar to that of the original decoder, while for Bose-Chaudhuri-Hocquenghem (BCH) codes, the performance slightly exceeds the original decoder.
Recently, deep learning has demonstrated improvements over classical decoding algorithms in various families of error-correcting codes with short to moderate block lengths. Neural decoders like neural belief propagation (NBP), constructed based on the Tanner graphs of linear codes, have been extensively studied, but their application to longer codes is limited due to increased network complexity with longer block lengths. This complexity leads to higher computational costs and deployment challenges, such as GPU memory usage. To address this, we propose a novel revolving framework for NBP-like decoders tailored to cyclic and QC-LDPC codes, a commonly used class of error correction codes. Our approach leverages the cyclic structure and section-wise cyclic structure inherent in cyclic and QC-LDPC codes respectively, significantly simplifying the complexity of the network. Experimental results demonstrate the effectiveness of our method across various cyclic and QC-LDPC codes. Especially, compared to the traditional decoding scheme based on the section-wise cyclic structure, our proposed decoder shows substantial improvements on 5G LDPC codes, exhibits better performance than the traditional min-sum decoder, and approaches the sum-product decoder without a noticeable error floor within the investigated noise levels.
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.
For short to moderate-length block codes, neural belief propagation (NBP) decoders are developed to ameliorate the performance of the belief propagation algorithm (BPA). The core concept underlying these decoders is to model belief propagation (BP) as a neural network, allowing adaptable decoding through trainable weights. The article proposes a novel Gated neural min-sum (GNMS) decoding technique with learnable parameters as a more hardware-efficient substitute for conventional NBP decoders. By using parameterised updates and gating techniques, our approach reformulates the min-sum (MS) algorithm, which is an approximation of BP, while preserving the flexibility of neural approaches and drastically lowering computing overhead. Additionally, we develop an autoencoder-assisted framework that supports flexible, learning-based optimisation for GNMS and Gated Neural Offset Min-Sum (GNOMS) decoders. The suggested decoders achieve error-correction performance at par with the most advanced NBP decoders by evaluating them on a variety of short to moderate-length block codes. Notably, when compared to traditional BP, the suggested approach lowers the bit error rate (BER) 10-fold for signal-to-noise ratio (SNR) greater than 3 dB. Furthermore, the efficacy of the proposed work is exhibited by comparing the computational complexity of the neural network-based decoders, highlighting the advantages in terms of practical implementation and hardware feasibility. The simulation results demonstrate that GNMS decoding with learnable parameters offers a convincing trade-off between complexity and performance, offering a potential path for future decoder design in contemporary communication systems.