text: Recently, it was shown that if multiplicative weights are assigned to the edges of a Tanner graph used in belief propagation decoding, it is possible to use deep learning techniques to find values for the weights which improve the error-correction performance of the decoder. Unfortunately, this approach requires many multiplications, which are generally expensive operations. In this paper, we suggest a more hardware-friendly approach in which offset min-sum decoding is augmented with learnable offset parameters. Our method uses no multiplications and has a parameter count less than half that of the multiplicative algorithm. This both speeds up training and provides a feasible path to hardware architectures. After describing our method, we compare the performance of the two neural decoding algorithms and show that our method achieves error-correction performance within 0.1 dB of the multiplicative approach and as much as 1 dB better than traditional belief propagation for the codes under consideration.
  @cite_5: This paper addresses the solution of bound-constrained optimization problems using algorithms that require only the availability of objective function values but no derivative information. We refer to these algorithms as derivative-free algorithms. Fueled by a growing number of applications in science and engineering, the development of derivative-free optimization algorithms has long been studied, and it has found renewed interest in recent time. Along with many derivative-free algorithms, many software implementations have also appeared. The paper presents a review of derivative-free algorithms, followed by a systematic comparison of 22 related implementations using a test set of 502 problems. The test bed includes convex and nonconvex problems, smooth as well as nonsmooth problems. The algorithms were tested under the same conditions and ranked under several criteria, including their ability to find near-global solutions for nonconvex problems, improve a given starting point, and refine a near-optimal solution. A total of 112,448 problem instances were solved. We find that the ability of all these solvers to obtain good solutions diminishes with increasing problem size. For the problems used in this study, TOMLAB MULTIMIN, TOMLAB GLCCLUSTER, MCS and TOMLAB LGO are better, on average, than other derivative-free solvers in terms of solution quality within 2,500 function evaluations. These global solvers outperform local solvers even for convex problems. Finally, TOMLAB OQNLP, NEWUOA, and TOMLAB MULTIMIN show superior performance in terms of refining a near-optimal solution.
  @cite_11: Two-dimensional (2-D) correction schemes are proposed to improve the performance of conventional min-sum (MS) decoding of irregular low density parity check codes. An iterative procedure based on parallel differential optimization is presented to obtain the optimal 2-D factors. Both density evolution analysis and simulation show that the proposed method provides a comparable performance as belief propagation (BP) decoding while requiring less complexity. Interestingly, the new method exhibits a lower error floor than that of BP decoding. With respect to conventional MS and 1-D normalized MS decodings, the 2-D normalized MS offers a better performance. The 2-D offset MS decoding exhibits a similar behavior.
summary: The approach given in @cite_11 is similar to NOMS in that it uses more than just one global offset. The main difference between this approach and NOMS is that the offsets found by @cite_11 do not depend on particular paths through the Tanner graph, and thus in general may have difficulty overcoming the effect of short cycles, especially if the code has only a few different check node degrees. For the BCH codes considered in this work, for example, every check node has the same degree, so the approach in @cite_11 would yield a single @math , thus reverting to OMS. NOMS, on the contrary, has the potential to improve performance even for regular codes. Finally, the genetic algorithm used to learn the offsets in @cite_11 is a random heuristic method, which, unlike gradient-based optimization methods, has been found to perform poorly as the size of the optimization problem increases @cite_5 .