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Asilomar Conference on Signals, Systems and Computers (ACSSC)

Complexity analysis of interior point methods for LP decoding

Yifan Sun, Lara Dolecek

Published Nov 1, 2011• 0 citations
Publisher

Abstract

Linear programming (LP) decoders can outperform currently used message-passing decoders in channel coding applications, but require prohibitively large complexity on even moderately sized codes. Previous works have proposed complexity-reducing algorithms that either relax the problem or modify the number of constraints; however, little work is done in optimizing solver implementation. We show that popular LP solvers like LIPSOL may not be efficient for LP decoding (LPD), and that an equivalent dual LP problem can be solved with equal accuracy but much more quickly. We propose an improved primal-dual method (iPD-MPC) whose overall runtime for both problem formulations outpreform LIPSOL. Additionally, as an alternative for memory-limited systems, we propose an improved hybrid gradient descent and Newton's method (iGD-NM) that further decreases overall runtime. In this way, we make LPD more feasible for channel codes of practical lengths.

DOI
10.1109/acssc.2011.6190085

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Linear programming (LP) decoders can outperform currently used message-passing decoders in channel coding applications, but require prohibitively large complexity on even moderately sized codes. Previous works have proposed complexity-reducing algorithms that either relax the problem or modify the number of constraints; however, little work is done in optimizing solver implementation. We show that popular LP solvers like LIPSOL may not be efficient for LP decoding (LPD), and that an equivalent dual LP problem can be solved with equal accuracy but much more quickly. We propose an improved primal-dual method (iPD-MPC) whose overall runtime for both problem formulations outpreform LIPSOL. Additionally, as an alternative for memory-limited systems, we propose an improved hybrid gradient descent and Newton's method (iGD-NM) that further decreases overall runtime. In this way, we make LPD more feasible for channel codes of practical lengths.
While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. By modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, we first prove that LP decoding can be performed by solving a number of LP problems that each contains at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate solving such systems iteratively. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.
While linear programming (LP) decoding provides more flexibility for finite-length performance analysis than iterative message-passing (IMP) decoding, it is computationally more complex to implement in its original form, due to both the large size of the relaxed LP problem, and the inefficiency of using general-purpose LP solvers. This paper explores ideas for fast LP decoding of low-density parity-check (LDPC) codes. We first prove, by modifying the previously reported Adaptive LP decoding scheme to allow removal of unnecessary constraints, that LP decoding can be performed by solving a number of LP problems that contain at most one linear constraint derived from each of the parity-check constraints. By exploiting this property, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithm is the solution of a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a preconditioning algorithm to facilitate iterative solution of such systems. The proposed preconditioning algorithm is similar to the encoding procedure of LDPC codes, and we demonstrate its effectiveness via both analytical methods and computer simulation results.
The linear programming (LP) decoding of Low-Density Parity-Check (LDPC) codes is widely concerned for its maximum likelihood (ML) features. It is that the ML decoding rule for LDPC codes can be relaxed to LP optimization problem. Therefore, this paper focus on an efficient algorithm called infeasible primal-dual interior-point (IPDIP) to solve the LP problem. In each iteration, the IPDIP algorithm obtains the predictor and corrector steps by solving the Karush-Kuhn-Tucker (KKT) equation twice. The predictor term is used to responsible for optimal solution and the corrector term keeps the current iteration away from the boundary of the feasible region. Furthermore, a modification of the centering parameter is developed to accelerate the convergence speed for the IPDIP algorithm. Simulation results of LP decoding demonstrate that the proposed IPDIP algorithm achieves beautiful bit error rate (BER) performance and good global convergence properties with less iteration number and time than other algorithms which only solve the KKT equation once by Newton method or use the not modified centering parameter.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has been derived for both binary and nonbinary decoding frameworks. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation for the nonbinary codes as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. The complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80, 48] LDPC code defined over ℤ4using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.
This paper explores ideas for fast linear programming (LP) decoding of low-density parity-check (LDPC) codes. We first propose a modification of adaptive LP decoding, and prove that it performs LP decoding by solving a number of linear programs that contain at most one linear constraint derived from each of the parity-check constraints. Then, as a step toward designing an efficient LP solver that exploits the structure of LDPC codes, we study a sparse interior-point implementation for solving this sequence of linear programs. Since the most complex part of each iteration of the interior-point algorithms is to solve a (usually ill-conditioned) system of linear equations for finding the step direction, we propose a framework for designing preconditioners to be used with the iterative methods for solving these systems. The effectiveness of the proposed approach is demonstrated via both analytical and simulation results.
Linear Programming (LP) decoding of Low-Density Parity-Check (LDPC) codes has attracted much attention in the research community in the past few years. The aim of LP decoding is to develop an algorithm which has error-correcting performance similar to that of the Sum-Product (SP) decoding algorithm, while at the same time it should be amenable to mathematical analysis. The LP decoding algorithm has also been extended to nonbinary linear codes by Flanagan et al. However, the most important problem with LP decoding for both binary and nonbinary linear codes is that the complexity of standard LP solvers such as the simplex algorithm remain prohibitively large for codes of moderate to large block length. To address this problem, Vontobel et al. proposed a low complexity LP decoding algorithm for binary linear codes which has complexity linear in the block length. In this paper, we extend the latter work and propose a low-complexity LP decoding algorithm for nonbinary linear codes. We use the LP formulation proposed by Flanagan et al. as a basis and derive a pair of primal-dual LP formulations. The dual LP is then used to develop the low-complexity LP decoding algorithm for nonbinary linear codes. In contrast to the binary low-complexity LP decoding algorithm, our proposed algorithm is not directly related to the nonbinary SP algorithm. Nevertheless, the complexity of the proposed algorithm is linear in the block length and is limited mainly by the maximum check node degree. As a proof of concept, we also present a simulation result for a [80,48][80,48][80,48] LDPC code defined over Z4\mathbb{Z}_4Z4​ using quaternary phase-shift keying over the AWGN channel, and we show that the error-correcting performance of the proposed LP decoding algorithm is similar to that of the standard LP decoding using the simplex solver.