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IEICE Transactions on Fundamentals of Electronics Communications and Computer Sciences

Bisection Method Assisted Affine Projection Algorithm in ADMM-LP Decoding of LDPC Codes

Rui Cheng, Yun JIANG, Qinglin Zhang, Qiaoqiao Xia

Published Jan 16, 2025• 0 citations
Publisher

Abstract

Many researchers have proposed optimization methods to reduce the computational complexity of the Euclidean projection onto check polytope in the alternating direction method of multipliers (ADMM) decoding for Low-Density Parity-Check (LDPC) codes. Existing the sparse affine projection algorithm (SAPA) projects the vector to be projected onto an χ-dimension affine hull and the dimension χ is fixed, resulting in deteriorating decoding performance. In this letter, bisection method assisted affine projection algorithm is proposed to determine the correct projection dimension for each the vector to be projected with the bisection method iterative algorithm. Simulation results show that the proposed algorithm can improve the accuracy of projection results by 68.2%. The FER performance of the proposed algorithm is almost the same as that of the exact projection algorithm, and compared with the sparse affine projection algorithm (SAPA), it can improve the FER performance by 0.14 dB as well as save average number of iterations by 3.2%.

DOI
10.1587/transfun.2024eal2109

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Many researchers have proposed optimization methods to reduce the computational complexity of the Euclidean projection onto check polytope in the alternating direction method of multipliers (ADMM) decoding for Low-Density Parity-Check (LDPC) codes. Existing the sparse affine projection algorithm (SAPA) projects the vector to be projected onto an χ-dimension affine hull and the dimension χ is fixed, resulting in deteriorating decoding performance. In this letter, bisection method assisted affine projection algorithm is proposed to determine the correct projection dimension for each the vector to be projected with the bisection method iterative algorithm. Simulation results show that the proposed algorithm can improve the accuracy of projection results by 68.2%. The FER performance of the proposed algorithm is almost the same as that of the exact projection algorithm, and compared with the sparse affine projection algorithm (SAPA), it can improve the FER performance by 0.14 dB as well as save average number of iterations by 3.2%.
We present a simplified alternating direction method of multipliers with linear programming (ADMM-LP) decoder for LDPC codes by developing a method of approximate projection onto the parity polytope. The algorithm projects onto the affine hull of theXXXclosest local codewords for each check of degree d, whereXXXcan be much smaller than d. We name this SAPA, the “sparse affine projection algorithm”. In contrast to exact projection, SAPA does not require a water-filling step and thus can be implemented with lower per-iteration complexity. We present numerical results which demonstrate not only that SAPA's performance, when used as part of an overall decoder, is close to that of exact projection, but also that the use of SAPA does not incur many additional iterations. This is in contrast to other approximate algorithms. Thus, SAPA is appropriate for use in limited-iteration decoders in high-throughput applications. Moreover, we analyze sparsity of polytope projections in the exact ADMM-LP decoding to explain the suitability of the proposed sparse approximation.
In this letter, a novel projection algorithm is proposed in which projection onto a triangle consisting of the three even-vertices closest to the vector to be projected replaces check polytope projection, achieving the same FER performance as exact projection algorithm in both high-iteration and low-iteration regime. Simulation results show that compared with the sparse affine projection algorithm (SAPA), it can improve the FER performance by 0.2dB as well as save average number of iterations by 4.3%.
Using linear programming (LP) decoding based on alternating direction method of multipliers (ADMM) for low-density parity-check (LDPC) codes shows lower complexity than the original LP decoding. However, the development of the ADMM-LP decoding algorithm could still be limited by the computational complexity of Euclidean projections onto parity check polytope. In this paper, we proposed a bisection method iterative algorithm (BMIA) for projection onto parity check polytope avoiding sorting operation and the complexity is linear. In addition, the convergence of the proposed algorithm is more than three times as fast as the existing algorithm, which can even be 10 times in the case of high input dimension.
Simplifying the Euclidean projection onto check polytope is an efficient way to reduce the computational complexity of alternating direction method of multipliers (ADMM) decoding algorithm for low-density parity-check (LDPC) codes. Existing algorithms for check polytope projection require sorting operation or iterative operation, which happens to be the most complex part of the projection. In this letter, a novel and fast projection algorithm is proposed without sorting and iterative operations. In the proposed algorithm, line segment projection replaces check polytope projection to approach approximate Euclidean projection at low computational complexity. Simulation results show that the proposed algorithm can substantially reduce the projection time while maintaining the frame error rate (FER) performance. In particular, the proposed algorithm can save the average projection time by 43% compared with cut search algorithm (CSA) when the dimension of the input vector is 20.
Alternating direction method of multipliers (ADMM) is a popular technique for linear-programming decoding of low-density parity-check codes. The computational complexity of ADMM is dominated by the Euclidean projection of a real-valued vector onto a parity-check polytope. Existing algorithms for such a projection all require sorting operations, which happen to be the most complex part of the projection. In this letter, we propose an iterative algorithm, without sorting operation, for projection onto the parity-check polytope. The proposed algorithm has a worst case complexity linear in the input dimension compared with the super-linear complexity of existing algorithms.
We present two strategies to reduce the complexity of the alternating direction method of multipliers with linear programming (ADMM-LP) decoder for LDPC codes. The complexity bottleneck of ADMM-LP decoding is projection onto the parity polytope. We propose the sparse affine projection algorithm (SAPA), which projects onto the affine hull of the χ closest local codewords for each check of degree d, where χ can be significantly smaller than d. Unlike exact projection, SAPA does not require a water-filling process, thus can be implemented with lower per-iteration complexity. Furthermore, we present a randomized layered scheduling framework for ADMM-LP decoding. In the randomized schedule, instead of updating all checks in a single iteration, more problematic checks are updated more often by using priors based the location of replica vectors inside or on the parity polytope. We demonstrate the decoder converges significantly faster with this randomized scheduling, thus it is well suited for limited-iteration applications.