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IEEE Information Theory Workshop (ITW)

Randomized Scheduling of ADMM-LP Decoding Based on Geometric Priors

Amirreza Asadzadeh, Masoud Barakatain, Jeebak Mitra, Frank R. Kschischang, Stark Christiaan Draper

Published Nov 1, 2022• 1 citation
Publisher

Abstract

We present a randomized schedule for alternating direction method of multipliers with linear programming (ADMM-LP) decoding of low-density parity check (LDPC) codes. The randomized schedule is based on horizontal layered decoding, where nodes are updated sequentially. Unlike existing layered decoding frameworks where all check nodes are updated exactly once per iteration, we propose a randomized schedule where more problematic nodes are updated more frequently. To do so, we sample from a probability mass function (PMF) over all check nodes and the check with sampled index is updated. The probability of each check to be updated is determined by its state. The PMF is constructed based on the distribution of replica (check) vectors inside or on the parity polytope. The randomized decoder usually converges faster than both standard and horizontal layered decoders, making it suitable for limited-iteration decoding in high-throughput applications.

DOI
10.1109/itw54588.2022.9965857

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We present a randomized schedule for alternating direction method of multipliers with linear programming (ADMM-LP) decoding of low-density parity check (LDPC) codes. The randomized schedule is based on horizontal layered decoding, where nodes are updated sequentially. Unlike existing layered decoding frameworks where all check nodes are updated exactly once per iteration, we propose a randomized schedule where more problematic nodes are updated more frequently. To do so, we sample from a probability mass function (PMF) over all check nodes and the check with sampled index is updated. The probability of each check to be updated is determined by its state. The PMF is constructed based on the distribution of replica (check) vectors inside or on the parity polytope. The randomized decoder usually converges faster than both standard and horizontal layered decoders, making it suitable for limited-iteration decoding in high-throughput applications.
We present two strategies to reduce the complexity of the alternating direction method of multipliers when applied to linear programming (ADMM-LP) decoding of low-density parity-check codes. First, to address the high complexity of computing a projection onto the parity polytope, the complexity bottleneck of ADMM-LP decoding, we propose the sparse affine projection algorithm (SAPA). SAPA projects onto the affine hull of χ ≤dnearby local codewords where the check degree isdand where χ can be significantly smaller thand. Unlike exact projection, SAPA does not require a water-filling process, and thus can be implemented with lower per-iteration complexity. Second, to reduce the number of effective iterations needed for ADMM-LP decoding, we propose a randomized layered scheduling framework. Rather than updating checks in round-robin fashion in each iteration, more “problematic” checks have a higher probability of being updated. The probability mass function that governs the selection of which checks to update is based upon the location of replica vectors inside (or on) the parity polytope. The resultant decoder converges significantly faster under this randomized scheduling than under round-robin scheduling. This makes it well suited for use in applications that limit the number of iterations.
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.
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.
For low-density parity-check (LDPC) codes, reducing the number of Euclidean projections, choosing a suitable scheduling strategy, and devising an improved penalty function are three effective ways to increase the alternating direction method of multipliers (ADMM) penalized decoding speed. By combining the above three ways, this paper proposes a horizontal layered scheduling ADMM penalized decoder based on the improved penalty function to speed up decoding. The experimental data indicate that the proposed decoder can obtain better performance and lower average number of iterations than the existing ADMM penalized decoders.
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.
The alternating direction method of multipliers (ADMM) approach to decoding low-density parity-check (LDPC) codes has garnered lots of attention due to its superior error correction performance in error floor regions. However, improving convergence characteristic of ADMM decoding efficiently remains to be a crucial but challenging issue for practical application. In order to tackle this challenge, in this paper we first propose an efficient Pro-ADMM-based decoding algorithm framework by inserting additional quadratic proximal terms to each iteration of ADMM decoding for stabilizing the variable updates. Second, an message-passing interpretation of Pro-ADMM decoder is considered, and then an improved version based on horizontal layered scheduling, named Pro-ADMM-HL decoder is developed to further enhance the convergence behavior of Pro-ADMM decoding. Additionally, we analyze that the complexity of the proposed Pro-ADMM-HL decoders at each iteration displays linearly to LDPC code length. Finally, the effectiveness of the proposed Pro-ADMM and Pro-ADMM-HL decoding algorithm in terms of error correction performance and convergence characteristic is demonstrated by simulation results.