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IEEE Communications Letters

Large Scale LP Decoding with Low Complexity

Guoqiang Zhang, Richard R. Heusdens, Willem Bastiaan Kleijn

Published Oct 17, 2013• 83 citations
Publisher

Abstract

Linear program (LP) decoding has become increasingly popular for error-correcting codes due to its simplicity and promising performance. Low-complexity and efficient iterative algorithms for LP decoding are of great importance for practical applications. In this paper we focus on solving the binary LP decoding problem by using the alternating direction method of multipliers (ADMM). Our main contribution is that we propose a linear-complexity algorithm for the projection onto a parity polytope (having a computational complexity of small O(d), where small d is the check-node degree), as compared to recent work , which has a computational complexity of small O(d log d). In particular, we show that the projection onto the parity polytope can be transformed to a projection onto a simplex.

DOI
10.1109/lcomm.2013.101413.130826

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Linear program (LP) decoding has become increasingly popular for error-correcting codes due to its simplicity and promising performance. Low-complexity and efficient iterative algorithms for LP decoding are of great importance for practical applications. In this paper we focus on solving the binary LP decoding problem by using the alternating direction method of multipliers (ADMM). Our main contribution is that we propose a linear-complexity algorithm for the projection onto a parity polytope (having a computational complexity of small O(d), where small d is the check-node degree), as compared to recent work , which has a computational complexity of small O(d log d). In particular, we show that the projection onto the parity polytope can be transformed to a projection onto a simplex.
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 (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.
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 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.
When binary linear error-correcting codes are used over symmetric channels, a relaxed version of the maximum likelihood decoding problem can be stated as a linear program (LP). This LP decoder can be used to decode error-correcting codes at bit-error-rates comparable to state-of-the-art belief propagation (BP) decoders, but with significantly stronger theoretical guarantees. However, LP decoding when implemented with standard LP solvers does not easily scale to the block lengths of modern error correcting codes. In this paper, we draw on decomposition methods from optimization theory, specifically the alternating direction method of multipliers (ADMM), to develop efficient distributed algorithms for LP decoding. The key enabling technical result is a “two-slice” characterization of the parity polytope, the polytope formed by taking the convex hull of all codewords of the single parity check code. This new characterization simplifies the representation of points in the polytope. Using this simplification, we develop an efficient algorithm for Euclidean norm projection onto the parity polytope. This projection is required by the ADMM decoder and its solution allows us to use LP decoding, with all its theoretical guarantees, to decode large-scale error correcting codes efficiently. We present numerical results for LDPC codes of lengths more than 1000. The waterfall region of LP decoding is seen to initiate at a slightly higher SNR than for sum-product BP, however an error floor is not observed for LP decoding, which is not the case for BP. Our implementation of LP decoding using the ADMM executes as fast as our baseline sum-product BP decoder, is fully parallelizable, and can be seen to implement a type of message-passing with a particularly simple schedule.