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IEEE Access

Joint-Way Compression for LDPC Neural Decoding Algorithm With Tensor-Ring Decomposition

Yuanhui Liang, Chan–Tong Lam, Benjamin K. Ng

Published Jan 1, 2023• 8 citations
Publisher

Abstract

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.

DOI
10.1109/access.2023.3252907

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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.
Compared with traditional low-density parity-check (LDPC) decoding algorithms, the current model-driven deep learning (DL)-based LDPC decoding algorithms face the disadvantage of high computational complexity. Based on the Neural Normalized Min-Sum (NNMS) algorithm, we propose a low-complexity model-driven DL-based LDPC decoding algorithm using Tensor-Train (TT) decomposition and syndrome loss function, called TT-NNMS+ algorithm. Our experiments show that the proposed TT-NNMS+ algorithm is more competitive than the NNMS algorithm in terms of bit error rate (BER) performance, memory requirement and computational complexity.
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.
Low-density parity-check (LDPC) codes are of prime importance in achieving near-Shannon capacity in current communication systems. However, the optimal decoding process for LDPC codes is computationally intensive. This paper presents a neural normalized min-sum (NNMS) decoding network that improves error correction capabilities with minimal computational overhead. A weight-sharing approach is adopted in the NNMS model, in which the correction factors (α, β) are shared across nodes within a single layer. This approach decreases the total number of parameters in the model compared to traditional neural decoders, making it easier for hardware implementation. The NNMS model is tested using a (576, 432) LDPC code for an additive white Gaussian noise channel with binary phase shift keying modulation. Simulation results show that the NNMS model outperforms traditional normalized min-sum (NMS) decoders. At a signal-to-noise ratio of 5 dB, the NNMS model has a bit-error rate (BER) of 1.0 × 10−7, whereas traditional NMS models have a less steep slope. In particular, the NNMS model has a coding gain of 1.25 dB compared to the traditional NMS model at a BER threshold of 1.0 × 10−3. This shows that the NNMS model is an efficient solution for high-performance, real-time digital communication systems.
Channel neural decoding is very promising as it outperforms the traditional channel decoding algorithms. Unfortunately, it still faces the disadvantage of high computational complexity and storage complexity compared with the traditional decoding algorithms. In this paper, we propose that low rank decomposition techniques based on tensor train decomposition and tensor ring decomposition can be utilized in neural offset min-sum (NOMS) and neural scale min-sim (NSMS) decoding algorithms. The experiment results show that the proposed two algorithms achieve near state-of-the-art performance with low complexity.
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.
Neural Normalized MinSum (N-NMS) decoding delivers better frame error rate (FER) performance on linear block codes than conventional Normalized MinSum (NMS) by assigning dynamic multiplicative weights to each check-to-variable node message in each iteration. Previous N-NMS efforts primarily investigated short block codes (N < 1000), because the number of N-NMS parameters required to be trained scales proportionately to the number of edges in the parity check matrix and the number of iterations. This imposes an impractical memory requirement for conventional tools such as Pytorch and Tensorflow to create the neural network and store gradients. This paper provides efficient methods of training the parameters of N-NMS decoders that support longer block lengths. Specifically, this paper introduces a family of Neural 2-dimensional Normalized (N-2D-NMS) decoders with various reduced parameter sets and shows how performance varies with the parameter set selected. The N-2D-NMS decoders share weights with respect to check node and/or variable node degree. Simulation results justify a reduced parameter set, showing that the trained weights of N- NMS have a smaller value for the neurons corresponding to larger check/variable node degree. Further simulation results on a (3096,1032) Protograph-Based Raptor-Like (PBRL) code show that the N-2D-NMS decoder can achieve the same FER as N- NMS while also providing at least a 99.7% parameter reduction. Furthermore, the N-2D-NMS decoder for the (16200,7200) DVBS- 2 standard LDPC code shows a lower error floor than belief propagation. Finally, this paper proposes a hybrid decoder training structure that utilizes a neural network which combines a feedforward module with a recurrent module. The decoding performance and parameter reduction of the hybrid training depends on the length of recurrent module of the neural network.