Compressed sensing (CS) demonstrates that sparse signals can be recovered from underdetermined linear measurements. We focus on the joint sparse recovery problem where multiple signals share the same common sparse support sets, and they are measured through the same sensing matrix. Leveraging a recent information theoretic characterization of single signal CS, we formulate the optimal minimum mean square error (MMSE) estimation problem, and derive a belief propagation algorithm, its relaxed version, for the joint sparse recovery problem and an approximate message passing algorithm. In addition, using density evolution, we provide a sufficient condition for exact recovery.
In this paper, we introduce a new support recovery algorithm from noisy measurements called Bayesian hypothesis test via belief propagation (BHT-BP). BHT-BP focuses on sparse support recovery rather than sparse signal estimation. The key idea behind BHT-BP is to detect the support set of a sparse vector using hypothesis test where the posterior densities used in the test are obtained by aid of belief propagation (BP). Since BP provides precise posterior information using the noise statistic, BHT-BP can recover the support with robustness against the measurement noise. In addition, BHT-BP has l…
It is shown how to combine ordered statistics decoding (OSD) with CRC-aided belief propagation list (CBPL) decoding of polar codes. Even when the reprocessing order of the OSD is as low as one, the new decoder is shown to significantly improve on CBPL. For reprocessing orders higher than one, we suggest partial reprocessing, where only error patterns associated with the least reliable part of the belief propagation decoded most reliable independent bits are considered. This type of partial reprocessing offers a trade-off between performance and computational complexity.
We propose a new algorithm for binary quantization based on the Belief Propagation algorithm with decimation over factor graphs of Low Density Generator Matrix (LDGM) codes. This algorithm, which we call Bias Propagation (BiP), can be considered as a special case of the Survey Propagation algorithm proposed for binary quantization by Wainwright et al. [8]. It achieves the same near-optimal rate-distortion performance with a substantially simpler framework and 10-100 times faster implementation. We thus challenge the widespread belief that binary quantization based on sparse linear codes cannot…
We consider the effect of log-likelihood ratio saturation on belief propagation decoder low-density parity-check codes. Saturation is commonly done in practice and is known to have a significant effect on error floor performance. Our focus is on threshold analysis and stability of density evolution. We analyze the decoder for standard low-density parity-check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation. Stability of density evolution is, on the other hand, rather strongly effected by saturation and the asymptotic qualitative effect of …
We consider the problem of identifying a pattern of faults from a set of noisy linear measurements. Unfortunately, maximum a posteriori probability estimation of the fault pattern is computationally intractable. To solve the fault identification problem, we propose a non-parametric belief propagation approach. We show empirically that our belief propagation solver is more accurate than recent state-of-the-art algorithms including interior point methods and semidefinite programming. Our superior performance is explained by the fact that we take into account both the binary nature of the individ…
We consider the effect of LLR saturation on belief propagation decoding of low-density parity-check codes. Saturation occurs universally in practice and is known to have a significant effect on error floor performance. Our focus is on threshold analysis and stability of density evolution. We analyze the decoder for certain low-density parity-check code ensembles and show that belief propagation decoding generally degrades gracefully with saturation. Stability of density evolution is, on the other hand, rather strongly affected by saturation and the asymptotic qualitative effect of saturation i…
This paper presents a novel propagation (BP) based decoding algorithm for polar codes. The proposed algorithm facilitates belief propagation by utilizing the specific constituent codes that exist in the factor graph, which results in an express journey (XJ) for belief information to propagate in each decoding iteration. In addition, this XJ-BP decoder employs a novel round-trip message passing scheduling method for the increased efficiency. The proposed method simplifies min-sum (MS) BP decoder by 40.6%. Along with the round-trip scheduling, the XJ-BP algorithm reduces the computational comple…
This paper proposes a low-computational Bayesian algorithm for noisy sparse recovery (NSR), called BHT-BP. In this framework, we consider an LDPC-like measurement matrices which has a tree-structured property, and additive white Gaussian noise. BHT-BP has a joint detection-and-estimation structure consisting of a sparse support detector and a nonzero estimator. The support detector is designed under the criterion of the minimum detection error probability using a nonparametric belief propagation (nBP) and composite binary hypothesis tests. The nonzeros are estimated in the sense of linear MMSE…
The recent work of Sommer, Feder and Shalvi presented a new family of codes called low density lattice codes (LDLC) that can be decoded efficiently and approach the capacity of the AWGN channel. A linear time iterative decoding scheme which is based on a message-passing formulation on a factor graph is given. In the current work we report our theoretical findings regarding the relation between the LDLC decoder and belief propagation. We show that the LDLC decoder is an instance of non-parametric belief propagation and further connect it to the Gaussian belief propagation algorithm. Our new res…
Permutation decoding gained recent interest as it can exploit the symmetries of a code in a parallel fashion. Moreover, it has been shown that by viewing permuted polar codes as polar subcodes, the set of usable permutations in permutation decoding can be increased. We extend this idea to pre-transformed polar codes, such as cyclic redundancy check (CRC)-aided polar codes, which previously could not be decoded using permutations due to their lack of automorphisms. Using belief propagation (BP)-based subdecoders, we showcase a performance close to CRC-aided SCL (CA-SCL) decoding. The proposed a…
This paper presents an enhanced belief propagation (BP) decoding algorithm and a reinforcement learning-based BP decoding algorithm for polar codes. The enhanced BP algorithm weighs each Processing Element (PE) input based on their signals and Euclidean distances using a heuristic metric. The proposed reinforcement learning-based BP decoding strategy relies on reweighting the messages and consists of two steps: we first weight each PE input based on their signals and Euclidean distances using a heuristic metric, then a Q-learning algorithm (QLBP) is employed to figure out the best correction f…
The equivalence of peeling decoding (PD) and Belief Propagation (BP) for low-density parity-check (LDPC) codes over the binary erasure channel is analyzed. Modifying the scheduling for PD, it is shown that exactly the same variable nodes (VNs) are resolved in every iteration than with BP. The decrease of erased VNs during the decoding process is analyzed instead of resolvable equations. This quantity can also be derived with density evolution, resulting in a drastic decrease in complexity. Finally, a scaling law using this quantity is established for spatially coupled LDPC codes.
Belief propagation is known to perform extremely well in many practical statistical inference and learning problems using graphical models, even in the presence of multiple loops. The iterative use of belief propagation algorithm on loopy graphs is referred to as Loopy Belief Propagation (LBP). Various sufficient conditions for convergence of LBP have been presented; however, general necessary conditions for its convergence to a unique fixed point remain unknown. Because the approximation of beliefs to true marginal probabilities has been shown to relate to the convergence of LBP, several meth…
We apply belief propagation (BP) to multi--user detection in a spread spectrum system, under the assumption of Gaussian symbols. We prove that BP is both convergent and allows to estimate the correct conditional expectation of the input symbols. It is therefore an optimal --minimum mean square error-- detection algorithm. This suggests the possibility of designing BP detection algorithms for more general systems. As a byproduct we rederive the Tse-Hanly formula for minimum mean square error without any recourse to random matrix theory.
This work describes a method of approximating matrix permanents efficiently using belief propagation. We formulate a probability distribution whose partition function is exactly the permanent, then use Bethe free energy to approximate this partition function. After deriving some speedups to standard belief propagation, the resulting algorithm requires $(n^2)$ time per iteration. Finally, we demonstrate the advantages of using this approximation.
Compressive sensing (CS) is a sampling technique designed for reducing the complexity of sparse data acquisition. One of the major obstacles for practical deployment of CS techniques is the signal reconstruction time and the high storage cost of random sensing matrices. We propose a new structured compressive sensing scheme, based on codes of graphs, that allows for a joint design of structured sensing matrices and logarithmic-complexity reconstruction algorithms. The compressive sensing matrices can be shown to offer asymptotically optimal performance when used in combination with Orthogonal …
Short survey about code on the graph by example of hardware friendly quasi-cycle LDPC code. We consider two main properties of code: weight enumerator (well known from classic code theory) and Trapping sets pseudocodewords weight spectrum (a subgraph in code graph's which become reasone of decoding failure under Belief propagation). In paper we show fast methods to measure first components of TS enumerator using EMD=ACE constrains for high girth codes on the graph using graph spectral classification method. This approach simplify solving trouble of agreed between minimal TS pseudocode weight a…
We investigate spatially coupled code ensembles. For transmission over the binary erasure channel, it was recently shown that spatial coupling increases the belief propagation threshold of the ensemble to essentially the maximum a-priori threshold of the underlying component ensemble. This explains why convolutional LDPC ensembles, originally introduced by Felstrom and Zigangirov, perform so well over this channel. We show that the equivalent result holds true for transmission over general binary-input memoryless output-symmetric channels. More precisely, given a desired error probability and …
In this paper a new message passing algorithm, which takes advantage of both tree-based re-parameterization and the knowledge of short cycles, is introduced for the purpose of decoding LDPC codes with short block lengths. The proposed algorithm is called variable factor appearance probability belief propagation (VFAP-BP) algorithm and is suitable for wireless communications applications, where both good decoding performance and low-latency are expected. Our simulation results show that the VFAP-BP algorithm outperforms the standard BP algorithm and requires a significantly smaller number of it…
We propose a fast real-time state estimator based on the belief propagation algorithm for the power system state estimation. The proposed estimator is easy to distribute and parallelize, thus alleviating computational limitations and allowing for processing measurements in real time. The presented algorithm may run as a continuous process, with each new measurement being seamlessly processed by the distributed state estimator. In contrast to the matrix-based state estimation methods, the belief propagation approach is robust to ill-conditioned scenarios caused by significant differences betwee…
We introduce a two-stage decimation process to improve the performance of neural belief propagation (NBP), recently introduced by Nachmani et al., for short low-density parity-check (LDPC) codes. In the first stage, we build a list by iterating between a conventional NBP decoder and guessing the least reliable bit. The second stage iterates between a conventional NBP decoder and learned decimation, where we use a neural network to decide the decimation value for each bit. For a (128,64) LDPC code, the proposed NBP with decimation outperforms NBP decoding by 0.75 dB and performs within 1 dB fro…
A low-density parity-check (LDPC) code is a linear block code described by a sparse parity-check matrix, which can be efficiently represented by a bipartite Tanner graph. The standard iterative decoding algorithm, known as belief propagation, passes messages along the edges of this Tanner graph. Density evolution is an efficient method to analyze the performance of the belief propagation decoding algorithm for a particular LDPC code ensemble, enabling the determination of a decoding threshold. The basic problem addressed in this work is how to optimize the Tanner graph so that the decoding thr…
In this work, we present a novel construction for solving the linear multiuser detection problem using the Gaussian Belief Propagation algorithm. Our algorithm yields an efficient, iterative and distributed implementation of the MMSE detector. We compare our algorithm's performance to a recent result and show an improved memory consumption, reduced computation steps and a reduction in the number of sent messages. We prove that recent work by Montanari et al. is an instance of our general algorithm, providing new convergence results for both algorithms.
We apply Guo and Wang's relaxed belief propagation (BP) method to the estimation of a random vector from linear measurements followed by a componentwise probabilistic measurement channel. Relaxed BP uses a Gaussian approximation in standard BP to obtain significant computational savings for dense measurement matrices. The main contribution of this paper is to extend the relaxed BP method and analysis to general (non-AWGN) output channels. Specifically, we present detailed equations for implementing relaxed BP for general channels and show that relaxed BP has an identical asymptotic large spars…
This paper considers the noisy group testing problem where among a large population of items some are defective. The goal is to identify all defective items by testing groups of items, with the minimum possible number of tests. The focus of this work is on the practical settings with a limited number of items rather than the asymptotic regime. In the current literature, belief propagation has been shown to be effective in recovering defective items from the test results. In this work, we adopt two variants of the belief propagation algorithm for the noisy group testing problem. These algorithm…
In this paper, we propose an alternative for routing based packet forwarding, which uses network coding to increase transmission efficiency, in terms of both compression and error resilience. This non-adaptive encoding is called quantized network coding, which involves random linear mapping in the real field, followed by quantization to cope with the finite capacity of the links. At the gateway node, which collects received quantized network coder packets, minimum mean squared error decoding is performed, by using belief propagation in the factor graph representation. Our simulation results sh…
In this paper, we propose a class of finite alphabet iterative decoder (FAID), called mutual information-maximizing quantized belief propagation (MIM-QBP) decoder, for decoding regular low-density parity-check (LDPC) codes. Our decoder follows the reconstruction-calculation-quantization (RCQ) decoding architecture that is widely used in FAIDs. We present the first complete and systematic design framework for the RCQ parameters, and prove that our design with sufficient precision at node update is able to maximize the mutual information between coded bits and exchanged messages. Simulation resu…
A sequential updating scheme (SUS) for belief propagation (BP) decoding of LDPC codes over Galois fields, $GF(q)$, and correlated Markov sources is proposed, and compared with the standard parallel updating scheme (PUS). A thorough experimental study of various transmission settings indicates that the convergence rate, in iterations, of the BP algorithm (and subsequently its complexity) for the SUS is about one half of that for the PUS, independent of the finite field size $q$. Moreover, this 1/2 factor appears regardless of the correlations of the source and the channel's noise model, while t…
Recently, Vontobel showed the relationship between Bethe free energy and annealed free energy for protograph factor graph ensembles. In this paper, annealed free energy of any random regular, irregular and Poisson factor graph ensembles are connected to Bethe free energy. The annealed free energy is expressed as the solution of maximization problem whose stationary condition equations coincide with equations of belief propagation since the contribution to partition function of particular type of variable and factor nodes has similar form of minus Bethe free energy. It gives simple derivation o…
We investigate an encoding scheme for lossy compression of a binary symmetric source based on simple spatially coupled Low-Density Generator-Matrix codes. The degree of the check nodes is regular and the one of code-bits is Poisson distributed with an average depending on the compression rate. The performance of a low complexity Belief Propagation Guided Decimation algorithm is excellent. The algorithmic rate-distortion curve approaches the optimal curve of the ensemble as the width of the coupling window grows. Moreover, as the check degree grows both curves approach the ultimate Shannon rate…
In this work, we show that polar belief propagation (BP) decoding exhibits an error floor behavior which is caused by clipping of the log-likelihood ratios (LLR). The error floor becomes more pronounced for clipping to smaller LLR-values. We introduce a single-value measure quantifying a "relative error floor", showing, by exhaustive simulations for different lengths, that the error floor is mainly caused by inadequate clipping values. We propose four modifications to the conventional BP decoding algorithm to mitigate this error floor behavior, demonstrating that the error floor is a decoder p…
The canonical problem of solving a system of linear equations arises in numerous contexts in information theory, communication theory, and related fields. In this contribution, we develop a solution based upon Gaussian belief propagation (GaBP) that does not involve direct matrix inversion. The iterative nature of our approach allows for a distributed message-passing implementation of the solution algorithm. We address the properties of the GaBP solver, including convergence, exactness, computational complexity, message-passing efficiency and its relation to classical solution methods. We use …
The decoding performance of polar codes strongly depends on the decoding algorithm used, while also the decoder throughput and its latency mainly depend on the decoding algorithm. In this work, we implement the powerful successive cancellation list (SCL) decoder on a GPU and identify the bottlenecks of this algorithm with respect to parallel computing and its difficulties. The inherent serial decoding property of the SCL algorithm naturally limits the achievable speed-up gains on GPUs when compared to CPU implementations. In order to increase the decoding throughput, we use a hybrid decoding s…
An information theoretic perspective on group testing problems has recently been proposed by Atia and Saligrama, in order to characterise the optimal number of tests. Their results hold in the noiseless case, where only false positives occur, and where only false negatives occur. We extend their results to a model containing both false positives and false negatives, developing simple information theoretic bounds on the number of tests required. Based on these bounds, we obtain an improved order of convergence in the case of false negatives only. Since these results are based on (computationall…
The canonical problem of solving a system of linear equations arises in numerous contexts in information theory, communication theory, and related fields. In this contribution, we develop a solution based upon Gaussian belief propagation (GaBP) that does not involve direct matrix inversion. The iterative nature of our approach allows for a distributed message-passing implementation of the solution algorithm. We also address some properties of the GaBP solver, including convergence, exactness, its max-product version and relation to classical solution methods. The application example of decorre…
We study the behavior of the belief-propagation (BP) algorithm affected by erroneous data exchange in a wireless sensor network (WSN). The WSN conducts a distributed binary hypothesis test where the joint statistical behavior of the sensor observations is modeled by a Markov random field whose parameters are used to build the BP messages exchanged between the sensing nodes. Through linearization of the BP message-update rule, we analyze the behavior of the resulting erroneous decision variables and derive closed-form relationships that describe the impact of stochastic errors on the performanc…
Compressive sensing (CS) is an emerging field based on the revelation that a small collection of linear projections of a sparse signal contains enough information for stable, sub-Nyquist signal acquisition. When a statistical characterization of the signal is available, Bayesian inference can complement conventional CS methods based on linear programming or greedy algorithms. We perform approximate Bayesian inference using belief propagation (BP) decoding, which represents the CS encoding matrix as a graphical model. Fast computation is obtained by reducing the size of the graphical model with…
In this paper, relaxed belief propagation (RBP) based detectors are proposed for multiple-input multiple-out (MIMO) system. The factor graph is leveraged to represent the MIMO channels, and based on which our algorithms are developed. Unlike the existing complicated standard belief propagation (SBP) detector that considers all the edges of the factor graph when updating messages, the proposed RBP focuses on partial edges, which largely reduces computational complexity. In particular, relax degree is introduced in to determine how many edges to be selected, whereby RBP is a generalized edge sel…
In this paper, we propose a solution to an AC state estimation problem in electric power systems using a fully distributed Gauss-Newton method. The proposed method is placed within the context of factor graphs and belief propagation algorithms and closed-form expressions for belief propagation messages exchanged along the factor graph are derived. The obtained algorithm provides the same solution as the conventional weighted least-squares state estimation. Using a simple example, we provide a step-by-step presentation of the proposed algorithm. Finally, we discuss the convergence behaviour usi…
The holographic transformation, belief propagation and loop calculus are generalized to problems in generalized probabilistic theories including quantum mechanics. In this work, the partition function of classical factor graph is represented by an inner product of two high-dimensional vectors both of which can be decomposed to tensor products of low-dimensional vectors. On the representation, the holographic transformation is clearly understood by using adjoint linear maps. Furthermore, on the formulation using inner product, the belief propagation is naturally defined from the derivation of t…
Quantum systems are the future candidates for computers and information processing devices. Information about quantum states and processes may be incomplete and scattered in these systems. We use a quantum version of Belief Propagation(BP) Algorithm to integrate the distributed information. In this algorithm the distributed information, which is in the form of density matrix, can be approximated to local structures. The validity of this algorithm is measured in comparison with Suzuki-Trotter(ST) method, using simulated information. ST in 3-body Heisenberg example gives a more accurate answer, …
We present a joint message passing approach that combines belief propagation and the mean field approximation. Our analysis is based on the region-based free energy approximation method proposed by Yedidia et al. We show that the message passing fixed-point equations obtained with this combination correspond to stationary points of a constrained region-based free energy approximation. Moreover, we present a convergent implementation of these message passing fixedpoint equations provided that the underlying factor graph fulfills certain technical conditions. In addition, we show how to include …
Loop Calculus introduced in [Chertkov, Chernyak '06] constitutes a new theoretical tool that explicitly expresses the symbol Maximum-A-Posteriori (MAP) solution of a general statistical inference problem via a solution of the Belief Propagation (BP) equations. This finding brought a new significance to the BP concept, which in the past was thought of as just a loop-free approximation. In this paper we continue a discussion of the Loop Calculus. We introduce an invariant formulation which allows to generalize the Loop Calculus approach to a q-are alphabet.
We present a novel distributed Gauss-Newton method for the non-linear state estimation (SE) model based on a probabilistic inference method called belief propagation (BP). The main novelty of our work comes from applying BP sequentially over a sequence of linear approximations of the SE model, akin to what is done by the Gauss-Newton method. The resulting iterative Gauss-Newton belief propagation (GN-BP) algorithm can be interpreted as a distributed Gauss-Newton method with the same accuracy as the centralized SE, however, introducing a number of advantages of the BP framework. The paper provi…
Near optimal decoding of good error control codes is generally a difficult task. However, for a certain type of (sufficiently) good codes an efficient decoding algorithm with near optimal performance exists. These codes are defined via a combination of constituent codes with low complexity trellis representations. Their decoding algorithm is an instance of (loopy) belief propagation and is based on an iterative transfer of constituent beliefs. The beliefs are thereby given by the symbol probabilities computed in the constituent trellises. Even though weak constituent codes are employed close t…
The belief propagation (BP) based algorithm is investigated as a potential decoder for both of error correcting codes and lossy compression, which are based on non-monotonic tree-like multilayer perceptron encoders. We discuss that whether the BP can give practical algorithms or not in these schemes. The BP implementations in those kind of fully connected networks unfortunately shows strong limitation, while the theoretical results seems a bit promising. Instead, it reveals it might have a rich and complex structure of the solution space via the BP-based algorithms.
The recent success in constructing asymptotically good quantum low-density parity-check (QLDPC) codes makes this family of codes a promising candidate for error-correcting schemes in quantum computing. However, conventional belief propagation (BP) decoding of QLDPC codes does not yield satisfying performance due to the presence of unavoidable short cycles in their Tanner graph and the special degeneracy phenomenon. In this work, we propose to decode QLDPC codes based on a check matrix with redundant rows, generated from linear combinations of the rows in the original check matrix. This approac…
Belief propagation (BP) is a classical algorithm that approximates the marginal distribution associated with a factor graph by passing messages between adjacent nodes in the graph. It gained popularity in the 1990's as a powerful decoding algorithm for LDPC codes. In 2016, Renes introduced a belief propagation with quantum messages (BPQM) and described how it could be used to decode classical codes defined by tree factor graphs that are sent over the classical-quantum pure-state channel. In this work, we propose an extension of BPQM to general binary-input symmetric classical-quantum (BSCQ) ch…
The performance of the generalized belief propagation algorithm for computing the noiseless capacity and mutual information rates of finite-size two-dimensional and three-dimensional run-length limited constraints is investigated. For each constraint, a method is proposed to choose the basic regions and to construct the region graph. Simulation results for the capacity of different constraints as a function of the size of the channel and mutual information rates of different constraints as a function of signal-to-noise ratio are reported. Convergence to the Shannon capacity is also discussed.
In this letter, we propose a joint active device detection and channel estimation framework based on factor graphs for asynchronous uplink grant-free massive multiple-antenna systems. We then develop the message-scheduling GAMP (MSGAMP) algorithm to perform joint active device detection and channel estimation. In MSGAMP we apply scheduling techniques based on the residual belief propagation (RBP) and the activity user detection (AUD) in which messages are generated using the latest available information. MSGAMP-type schemes show a good performance in terms of activity error rate and normalized…
We introduce the framework of cooperative simultaneous localization and tracking (CoSLAT), which provides a consistent combination of cooperative self-localization (CSL) and distributed target tracking (DTT) in sensor networks without a fusion center. CoSLAT extends simultaneous localization and tracking (SLAT) in that it uses also intersensor measurements. Starting from a factor graph formulation of the CoSLAT problem, we develop a particle-based, distributed message passing algorithm for CoSLAT that combines nonparametric belief propagation with the likelihood consensus scheme. The proposed …
The group testing problem asks for efficient pooling schemes and algorithms that allow to screen moderately large numbers of samples for rare infections. The goal is to accurately identify the infected samples while conducting the least possible number of tests. Exploring the use of techniques centred around the Belief Propagation message passing algorithm, we suggest a new test design that significantly increases the accuracy of the results. The new design comes with Belief Propagation as an efficient inference algorithm. Aiming for results on practical rather than asymptotic problem sizes, w…
Belief propagation is a powerful tool in statistical physics, machine learning, and modern coding theory. As a decoding method, it is ubiquitous in classical error correction and has also been applied to stabilizer-based quantum error correction. The algorithm works by passing messages between nodes of the factor graph associated with the code and enables efficient decoding, in some cases even up to the Shannon capacity of the channel. Here we construct a belief propagation algorithm which passes quantum messages on the factor graph and is capable of decoding the classical-quantum channel with…
Quantum error correction is necessary to protect logical quantum states and operations. However, no meaningful data protection can be made when the syndrome extraction is erroneous due to faulty measurement gates. Quantum data-syndrome (DS) codes are designed to protect the data qubits and syndrome bits concurrently. In this paper, we propose an efficient decoding algorithm for quantum DS codes with sparse check matrices. Based on a refined belief propagation (BP) decoding for stabilizer codes, we propose a DS-BP algorithm to handle the quaternary quantum data errors and binary syndrome bit er…
