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International Conference on Quantum Technologies, Communications, Computing, Hardware and Embedded Systems Security (iQ-CCHESS)

A Concise Polytope Projection Method for ADMM-Assisted LP Decoding of LDPC Codes

Parul Parul, Manish Gupta, Rajesh Kumar Narula, Gurinderjit Kaur

Published Sep 15, 2023• 0 citations
Publisher

Abstract

The alternating direction method of multipliers (ADMM) has proven to be a highly efficient method for implementing linear programming (LP) decoding for low-density parity-check (LDPC) codes. The projection on the parity polytope is the most time-intensive operation involved in the ADMM-assisted LP decoding. The existing projection methods involve sorting operations or complicated iterations which are difficult to perform by hardware systems. In this paper, an efficient parity-polytope projection algorithm is proposed by mapping the projection problem onto an isomorphic problem of probability simplex. The projection on probability simplex is obtained by utilizing its recurrent structure. This algorithm does not require sorting operations. Each iteration fixes at least one component of the input vector and has straightforward iteration steps. The simulation results show a noticeable reduction in the processing time for obtaining the projection of a given vector. Due to these advantages, it is beneficial for the effective implementation of ADMM-assisted LP decoding systems.

DOI
10.1109/iq-cchess56596.2023.10391405

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The alternating direction method of multipliers (ADMM) has proven to be a highly efficient method for implementing linear programming (LP) decoding for low-density parity-check (LDPC) codes. The projection on the parity polytope is the most time-intensive operation involved in the ADMM-assisted LP decoding. The existing projection methods involve sorting operations or complicated iterations which are difficult to perform by hardware systems. In this paper, an efficient parity-polytope projection algorithm is proposed by mapping the projection problem onto an isomorphic problem of probability simplex. The projection on probability simplex is obtained by utilizing its recurrent structure. This algorithm does not require sorting operations. Each iteration fixes at least one component of the input vector and has straightforward iteration steps. The simulation results show a noticeable reduction in the processing time for obtaining the projection of a given vector. Due to these advantages, it is beneficial for the effective implementation of ADMM-assisted LP decoding systems.
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.
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.
The alternating direction method of multipliers has recently been adapted for linear programming decoding of low-density parity-check codes. The computation of the projection onto the parity polytope is the core of this algorithm and usually involves a sorting operation, which is the main effort of the projection. In this paper, we present an algorithm with low complexity to compute this projection. The algorithm relies on new findings in the recursive structure of the parity polytope and iteratively fixes selected components. As shown in our realistic simulation setup, it requires up to 37% less arithmetical operations compared with state-of-the-art projections. Additionally, it does not involve a sorting operation, which is needed in all exact state-of-the-art projection algorithms. These two benefits make it appealing for efficient hardware and software implementations.
Alternating Direction Method of Multipliers (ADM-M) is an effective method for Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes. In the ADMM-based LDPC decoding algorithm, Euclidean projection on the check polytope is the key step. Current Euclidean projection algorithms in general require either sorting or iterative operations. However, when decoding LDPC codes with high coding rate, the sorting operation brings a high level of computation complexity. Moreover, although the current iterative projection algorithms do not involve the complex operation of sorting, its convergence speed is slow. In this paper, based on trend extrapolation, a reduced-complexity iterative check polytope projection algorithm is proposed. In the proposed algorithm, the target projection point can be found based on the variation trend of the geometric position of the projection points obtained from previous iterations. Therefore, it ensures fast convergence and does not sacrifice on bit error rate (BER) performance. The simulation results confirm that the proposed algorithm can effectively reduce the decoding complexity, especially for LDPC codes with high coding rate.
In this letter, we develop an efficient linear programming (LP) decoding algorithm for low-density parity-check (LDPC) codes. The LP relaxation is formulated based on a check-node decomposition approach. Our main contributions are as follows. First, we propose an algorithm based on the alternating direction method of multipliers (ADMM) technique to solve this LP relaxation. By exploiting the orthogonality structure of the LP model, each ADMM update step can be implemented in parallel. Second, the proposed decoding algorithm under this LP formulation eliminates the Euclidean projection on the check polytope compared with the existing ADMM-based LP decoding algorithms. Third, the feasibility analysis of the proposed algorithm is presented. Moveover, complexity analysis shows that our proposed LP decoder in each iteration has a lower complexity than the state-of-the-art ADMM-based LP decoders. Simulation results demonstrate that the proposed LP decoder achieves better performance than other competing ADMM-based LP decoders in terms of decoding time.
The Euclidean projection onto check polytopes is the most time-consuming operation in the linear programming (LP) decoding based on alternating direction method of multipliers (ADMM) for low-density parity-check (LDPC) codes. In this letter, instead of reducing the complexity of Euclidean projection itself, we propose a new method to reduce the decoding complexity of ADMM-based LP decoder by decreasing the number of Euclidean projections. In particular, if all absolute values of the element-wise differences between the input vector of Euclidean projection in the current iteration and that in the previous iteration are less than a predefined value, then the Euclidean projection at the current iteration will be no longer performed. Simulation results show that the proposed decoder can still save roughly 20% decoding time even if both the over-relaxation and early termination techniques are used.