This paper presents a Deep Autoencoder–LDPC–OFDM (DAE–LDPC–OFDM) transceiver architecture that integrates a learned belief propagation (BP) decoder to achieve robust, energy-efficient, and adaptive wireless communication. Unlike conventional modular systems that treat encoding, modulation, and decoding as independent stages, the proposed framework performs end-to-end joint optimization of all components, enabling dynamic adaptation to varying channel and noise conditions. The learned BP decoder introduces trainable parameters into the iterative message-passing process, allowing adaptive refinement of log-likelihood ratio (LLR) statistics and enhancing decoding accuracy across diverse SNR regimes. Extensive experimental results across multiple datasets and channel scenarios demonstrate the effectiveness of the proposed design. At 10 dB SNR, the DAE–LDPC–OFDM achieves a BER of 1.72% and BLER of 2.95%, outperforming state-of-the-art models such as Transformer–OFDM, CNN–OFDM, and GRU–OFDM by 25–30%, and surpassing traditional LDPC–OFDM systems by 38–42% across all tested datasets. The system also achieves a PAPR reduction of 26.6%, improving transmitter power amplifier efficiency, and maintains a low inference latency of 3.9 ms per frame, validating its suitability for real-time applications. Moreover, it maintains reliable performance under time-varying, interference-rich, and multipath fading channels, confirming its robustness in realistic wireless environments. The results establish the DAE–LDPC–OFDM as a high-performance, power-efficient, and scalable architecture capable of supporting the demands of 6G and beyond, delivering superior reliability, low-latency performance, and energy-efficient communication in next-generation intelligent networks.

As quantum hardware advances toward enabling error-corrected quantum circuits in the near future, the absence of an efficient polynomial-time decoding algorithm for logical circuits presents a critical bottleneck. While quantum memory decoding has been well studied, inevitable correlated errors introduced by transversal entangling logical gates prevent the straightforward generalization of quantum memory decoders. Here we introduce a data-centric, modular decoder framework, the Multi-Core Circuit Decoder (MCCD), which consists of decoder modules corresponding to each logical operation supported by the quantum hardware. The MCCD handles both single-qubit and entangling gates within a unified framework. We train MCCD using mirror-symmetric random Clifford circuits, demonstrating its ability to effectively learn correlated decoding patterns. Through extensive testing on circuits substantially deeper than those used in training, we show that MCCD maintains high logical accuracy while exhibiting competitive polynomial decoding time across increasing circuit depths and code distances. When compared with conventional decoders such as minimum weight perfect matching (MWPM), most likely error (MLE) and belief propagation with ordered statistics post-processing (BP-OSD), MCCD achieves competitive accuracy with substantially better time efficiency, particularly for circuits with entangling gates. Our approach provides a noise-model-agnostic solution to the decoding challenge in deep logical quantum circuits.

Distributed kernel mean embedding Gaussian belief propagation for underwater multi-sensor multi-target passive tracking

Belief Propagation (BP) is a fundamental heuristic for solving Constraint Optimization Problems (COPs), yet its practical applicability is constrained by slow convergence and instability in loopy factor graphs. While Damped BP (DBP) improves convergence by using manually tuned damping factors, its reliance on labor-intensive hyperparameter optimization limits scalability. Deep Attentive BP (DABP) addresses this by automating damping through recurrent neural networks (RNNs), but introduces significant memory overhead and sequential computation bottlenecks. To reduce memory usage and accelerate deep belief propagation, this paper introduces Fast Deep Belief Propagation (FDBP), a deep learning framework that improves COP solving through online self-supervised learning and graphics processing unit (GPU) acceleration. FDBP decouples the learning of damping factors from BP message passing, inferring all parameters for an entire BP iteration in a single step, and leverages mixed precision to further optimize GPU memory usage. This approach substantially improves both the efficiency and scalability of BP optimization. Extensive evaluations on synthetic and real-world benchmarks highlight the superiority of FDBP, especially for large-scale instances where DABP fails due to memory constraints. Moreover, FDBP achieves an average speedup of 2.87× over DABP with the same restart counts. Because BP for COPs is a mathematically grounded GPU-parallel message-passing framework that bridges applied mathematics, computing, and machine learning, and is widely applicable across science and engineering, our work offers a promising step toward more efficient solutions to these problems.

With the rapid growth of data volume in sensor networks, lossy source coding systems achieve high-efficiency data compression with low distortion under limited transmission bandwidth. However, conventional compression algorithms rely on a two-stage framework with high computational complexity and frequently struggle to balance compression performance with generalization ability. To address these issues, an end-to-end lossy compression method is proposed in this paper. The approach integrates an enhanced belief propagation algorithm with a multi-layer perceptron neural network, aiming to introduce a novel joint optimization architecture described as "encoding-structured encoding-decoding". In addition, a quantization module incorporating random perturbation and the straight-through estimator is designed to address the non-differentiability in the quantization process. Simulation results demonstrate that the proposed system significantly improves compression performance while offering superior generalization and reconstruction quality. Furthermore, the designed neural architecture is both simple and efficient, reducing system complexity and enhancing feasibility for practical deployment.

For the tensor principal component analysis (tensor PCA) problem, we propose a new hierarchy of increasingly powerful algorithms with increasing runtime. Our hierarchy is analogous to the sum-of-squares (SOS) hierarchy but is instead inspired by statistical physics and related algorithms such as belief propagation and AMP (approximate message passing). Our level-ℓ algorithm can be thought of as a linearized message-passing algorithm that keeps track of ℓ-wise dependencies among the hidden variables. Specifically, our algorithms are spectral methods based on the Kikuchi Hessian , which generalizes the well-studied Bethe Hessian to the higher-order Kikuchi free energies. It is known that AMP, the flagship algorithm of statistical physics, has substantially worse performance than SOS for tensor PCA. In this work, we ‘redeem’ the statistical physics approach by showing that our hierarchy gives a polynomial-time algorithm matching the performance of SOS. Our hierarchy also yields a continuum of subexponential-time algorithms, and we prove that these achieve the same (conjecturally optimal) tradeoff between runtime and statistical power as SOS. Our proofs are much simpler than prior work, and also apply to the related problem of refuting random k -XOR formulas. The results we present here apply to tensor PCA for tensors of all orders, and to k -XOR when k is even. Our methods suggest a new avenue for systematically obtaining optimal algorithms for Bayesian inference problems, and our results constitute a step toward unifying the statistical physics and sum-of-squares approaches to algorithm design.

We present a new decoder for the surface code, which combines the accuracy of the tensor-network decoders with the efficiency and parallelism of the belief-propagation algorithm. Our main idea is to replace the expensive tensor-network contraction step in the tensor-network decoders with the algorithm—a recent approximate contraction algorithm, based on belief propagation. Our decoder is therefore a belief-propagation decoder that works in the degenerate maximal likelihood decoding framework. Unlike conventional tensor-network decoders, our algorithm can run efficiently in parallel, and may therefore be suitable for real-time decoding. We numerically test our decoder and show that for a large range of lattice sizes and noise levels it delivers a logical error probability that outperforms the Minimal Weight Perfect Matching decoder, sometimes by more than an order of magnitude.

ABSTRACT Machine Learning (ML) has proven to be an effective tool for optimizing several aspects of channel codes in communication systems. In this work, ML-based schemes have been adapted to the 5G New Radio (NR) Low-Density Parity Check (LDPC) codes to help them meet the stringent requirements of ultra-reliable low-latency communications (URLLC) in 5G NR. Furthermore, enhancements to the modulation schemes have also been applied to attempt to outperform the complex and robust Layered Belief Propagation Algorithm (LBPA) using the lower complexity Normalized Min-Sum Algorithm (NMSA) decoder. By combining the ML-based schemes and the enhanced modulation schemes, NMSA effectively provided E b / N 0 gains of up to 1 dB and latency reductions of up to 0.53 ms compared to the conventional Layered Belief Propagation Algorithm.

The code distance is a critical parameter of quantum stabilizer codes (QSCs), and determining it—whether exactly or approximately—is known to be an NP-complete problem. However, its upper bound can be determined efficiently by some methods such as the Monte Carlo method. Leveraging the Monte Carlo method, we propose an algorithm to compute the upper bound on the code distance of a given QSC using fully decoupled belief propagation combined with ordered statistics decoding (FDBP-OSD). Our algorithm demonstrates high precision: for various QSCs with known distances, the computed upper bounds match the actual values. Additionally, we explore upper bounds for the minimum weight of logical X operators in the Z-type Tanner-graph-recursive-expansion (Z-TGRE) code and the Chamon code—an XYZ product code constructed from three repetition codes. The results on Z-TGRE codes align with theoretical analysis, while the results on Chamon codes suggest that XYZ product codes may achieve a code distance of , which supports the conjecture of Leverrier et al.

Enhancing grammatical documentation for endangered languages with graph-based meaning representation and Loopy Belief Propagation

In modern communication systems, the concatenation of a low-density parity-check (LDPC) code with a cyclic redundancy check (CRC) code is commonly used for error correction. In this paper, we propose a low-complexity two-stage scheme for decoding these codes using their concatenation structures. In the first stage, the traditional belief propagation (BP)-based iterative algorithm with a relative small maximum number of iterations is performed for decoding the LDPC code. If an LDPC codeword is obtained in this stage, the decoding process terminates. Otherwise, the second stage of the decoding process is performed, in which the guessing random additive noise decoding (GRAND) algorithm is applied to the CRC code. A list of information sequences satisfying the CRC check is obtained, each of which is then encoded to an LDPC codeword. The most likely codeword among them is the output of the decoding approach. The simulation results indicate that the proposed two-stage decoding approach can outperform the traditional BP-based iterative algorithm with a large maximum number of iterations. Moreover, the average complexity of the proposed approach is relatively low.

For short to moderate-length block codes, neural belief propagation (NBP) decoders are developed to ameliorate the performance of the belief propagation algorithm (BPA). The core concept underlying these decoders is to model belief propagation (BP) as a neural network, allowing adaptable decoding through trainable weights. The article proposes a novel Gated neural min-sum (GNMS) decoding technique with learnable parameters as a more hardware-efficient substitute for conventional NBP decoders. By using parameterised updates and gating techniques, our approach reformulates the min-sum (MS) algorithm, which is an approximation of BP, while preserving the flexibility of neural approaches and drastically lowering computing overhead. Additionally, we develop an autoencoder-assisted framework that supports flexible, learning-based optimisation for GNMS and Gated Neural Offset Min-Sum (GNOMS) decoders. The suggested decoders achieve error-correction performance at par with the most advanced NBP decoders by evaluating them on a variety of short to moderate-length block codes. Notably, when compared to traditional BP, the suggested approach lowers the bit error rate (BER) 10-fold for signal-to-noise ratio (SNR) greater than 3 dB. Furthermore, the efficacy of the proposed work is exhibited by comparing the computational complexity of the neural network-based decoders, highlighting the advantages in terms of practical implementation and hardware feasibility. The simulation results demonstrate that GNMS decoding with learnable parameters offers a convincing trade-off between complexity and performance, offering a potential path for future decoder design in contemporary communication systems.

Abstract To improve the decoding performance of quantum error-correcting codes in asymmetric noise channels, a neural network-based decoding algorithm for bias-tailored quantum codes is proposed. The algorithm consists of a biased noise model, a neural belief propagation decoder, a convolutional optimization layer, and a multi-objective loss function. The biased noise model simulates asymmetric error generation, providing a training dataset for decoding. The neural network, leveraging dynamic weight learning and a multi-objective loss function, mitigates error degeneracy. Additionally, the convolutional optimization layer enhances early-stage convergence efficiency. Numerical results show that for bias-tailored quantum codes, our decoder performs much better than the belief propagation (BP) with ordered statistics decoding (BP + OSD). Our decoder achieves an order of magnitude improvement in the error suppression compared to higher-order BP + OSD. Furthermore, the decoding threshold of our decoder for surface codes reaches a high threshold of 20%.

A PEXIT Analysis of Belief Propagation Polar Decoder with Approximated LLR

A Hybrid Inference Architecture Incorporating Neural Network With Belief Propagation for AI Receivers

Weighted belief propagation (WBP) for the decoding of linear block codes is considered. In WBP, the Tanner graph of the code is unrolled with respect to the iterations of the belief propagation decoder. Then, weights are assigned to the edges of the resulting recurrent network and optimized offline using a training dataset. The main contribution of this paper is an adaptive WBP where the weights of the decoder are determined for each received word. Two variants of this decoder are investigated. In the parallel WBP decoders, the weights take values in a discrete set. A number of WBP decoders are run in parallel to search for the best sequence- of weights in real time. In the two-stage decoder, a small neural network is used to dynamically determine the weights of the WBP decoder for each received word. The proposed adaptive decoders demonstrate significant improvements over the static counterparts in two applications. In the first application, Bose–Chaudhuri–Hocquenghem, polar and quasi-cyclic low-density parity-check (QC-LDPC) codes are used over an additive white Gaussian noise channel. The results indicate that the adaptive WBP achieves bit error rates (BERs) up to an order of magnitude less than the BERs of the static WBP at about the same decoding complexity, depending on the code, its rate, and the signal-to-noise ratio. The second application is a concatenated code designed for a long-haul nonlinear optical fiber channel where the inner code is a QC-LDPC code and the outer code is a spatially coupled LDPC code. In this case, the inner code is decoded using an adaptive WBP, while the outer code is decoded using the sliding window decoder and static belief propagation. The results show that the adaptive WBP provides a coding gain of 0.8 dB compared to the neural normalized min-sum decoder, with about the same computational complexity and decoding latency.

This paper considers the distributed group target tracking (DGTT) problem under sensors with limited and different field of views (FoVs). Usually, for the tracking of groups, targets within groups are closely spaced and move in a coordinated manner. These groups can split or merge, and the numbers of targets in groups may be large, which lead to more challenging issues related to data association, filtering and computational complexities. Particularly, these challenges may be further complicated in distributed fusion system architectures. To deal with these difficulties, we propose a consensus-based DGTT method within the belief propagation (BP) framework, which introduces undetected targets inside the FoV or new targets outside the FoV and performs the probabilistic track association via BP. Meanwhile, the obtained track association probabilities make it possible to exploit a probabilistic consensus fusion scheme for fusing local target densities. Furthermore, the proposed method exhibits computational scalability scaling only linearly on the numbers of group partitions, local measurements and neighboring sensors, and scaling quadratically on the number of targets. Numerical results validate the performance of the proposed method.

We advance a mathematical framework for collective conviction by deriving a continuum theory from the network-based model introduced by us recently. The resulting equation governs the evolution of belief through a degenerate anisotropic logistic–diffusion process, where diffusion slows as conviction saturates. In one spatial dimension, we prove global well-posedness, demonstrate spectral front pinning that arrests the spread of influence at finite depth, and construct explicit traveling-wave solutions. In two dimensions, we uncover a geometric mechanism of curvature–induced quenching, where belief propagation halts along regions of low effective mobility and curvature. Building on this insight, we formulate a variational principle for optimal control under resource constraints. The derived feedback law prescribes how to spatially allocate repression effort to maximize inhibition of front motion, concentrating resources along high-curvature, low-mobility arcs. Numerical simulations validate the theory, illustrating how localized suppression dramatically reduces transverse spread without affecting fast axes. These results bridge analytical modeling with societal phenomena such as protest diffusion, misinformation spread, and institutional resistance, offering a principled foundation for selective intervention policies in structured populations.

Gaussian belief propagation for dynamic obstacle avoidance and formation control in second-order multi-agent systems

BackgroundMontelukast(MTK) is a leukotriene receptor antagonist widely used clinically for treating asthma and rhinitis. However, many adverse events (AEs) have been reported. In this study, we aimed to investigate MTK’s adverse drug reactions (ADRs) using real data from the U.S. Food and Drug Administration Adverse Event Reporting System (FAERS) database.MethodsWe assessed the disproportionality of MTK-associated AEs by calculating metrics such as the ratio of reported ratios (ROR), proportional reporting ratios (PRR), Bayesian Belief Propagation Neural Networks (BCPNN), and Gamma-Poisson Shrinkers (GPS).ResultsFrom 2004 to the third quarter of 2023, of the 20,340,254 case reports in the FAERS database, 86,732 MTK reports were recorded as “principal suspect (PS)” AEs.Disproportionate analyses identified 431 preferred terms (PTs) associated with MTK in 27 organ systems. Unexpected major AEs were noted, such as Suffocation feeling, Adrenal suppression, Sudden visual loss and Endocardial fibrosis, none of which were mentioned in the drug insert.ConclusionOur findings are consistent with clinical observations highlighting potential new and unexpected ADR signals associated with MTK. Further prospective clinical studies are needed to confirm and elucidate the relationship between MTK and these ADRs. This study provides a fresh and unique perspective on the study of adverse drug events.