On the Effective Weights of Pseudocodewords for Codes Defined on Graphs with Cycles

Forney, G. David; Koetter, R.; Kschischang, Frank R.; Reznik, Alex

doi:10.1007/978-1-4613-0165-3_5

Cited by 96 publications

(167 citation statements)

References 8 publications

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“…Not surprisingly, given the role of the constraint set L(G) in the Bethe variational problem, there are close connections between LP decoding and standard iterative algorithms like sum-product decoding [76,78,136]. Among other connections, the fractional extreme points of the first-order LP relaxation have a very specific interpretation as pseudocodewords of the underlying code, studied in earlier work on iterative decoding [83,86,257]. ; it is easy to verify that it satisfies the box inequalities and the forbidden set constraints (8.25) that define the LP relaxation.…”

Section: Example 83 (Lp Relaxations For Error-control Coding)mentioning

confidence: 99%

Graphical Models, Exponential Families, and Variational Inference

Wainwright¹,

Jordan²

2007

FNT in Machine Learning

2,184

2,452

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The formalism of probabilistic graphical models provides a unifying framework for capturing complex dependencies among random variables, and building large-scale multivariate statistical models. Graphical models have become a focus of research in many statistical, computational and mathematical fields, including bioinformatics, communication theory, statistical physics, combinatorial optimization, signal and image processing, information retrieval and statistical machine learning. Many problems that arise in specific instancesincluding the key problems of computing marginals and modes of probability distributions -are best studied in the general setting. Working with exponential family representations, and exploiting the conjugate duality between the cumulant function and the entropy for exponential families, we develop general variational representations of the problems of computing likelihoods, marginal probabilities and most probable configurations. We describe how a wide variety of algorithms -among them sum-product, cluster variational methods, expectation-propagation, mean field methods, max-product and linear programming relaxation, as well as conic programming relaxations -can all be understood in terms of exact or approximate forms of these variational representations. The variational approach provides a complementary alternative to Markov chain Monte Carlo as a general source of approximation methods for inference in large-scale statistical models.

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Section: Example 83 (Lp Relaxations For Error-control Coding)mentioning

confidence: 99%

Graphical Models, Exponential Families, and Variational Inference

Wainwright¹,

Jordan²

2007

FNT in Machine Learning

2,184

2,452

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“…In this section, we provide some discussions based on pseudocodewords [13], [14], [15] that help us understand Algorithm D. We first briefly provide some definitions and results from [15]. For more details, readers are referred to [15].…”

Section: Application Of Pseudo-codewords To the Analysis Of Algormentioning

confidence: 99%

Results on Punctured Low-Density Parity-Check Codes and Improved Iterative Decoding Techniques

Pishro-Nik

Fekri

2007

IEEE Trans. Inform. Theory

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Abstract-This paper first introduces an improved decoding algorithm for low-density parity-check (LDPC) codes over binaryinput-output-symmetric memoryless channels. Then some fundamental properties of punctured LDPC codes are presented. It is proved that for any ensemble of LDPC codes, there exists a puncturing threshold. It is then proved that for any rates R1 and R2 satisfying 0 < R1 < R2 < 1, there exists an ensemble of LDPC codes with the following property. The ensemble can be punctured from rate R1 to R2 resulting in asymptotically good codes for all rates R1 R R2. Specifically, this implies that rates arbitrarily close to one are achievable via puncturing. Bounds on the performance of punctured LDPC codes are also presented. It is also shown that punctured LDPC codes are as good as ordinary LDPC codes. For BEC and arbitrary positive numbers R 1 < R 2 < 1, the existence of the sequences of punctured LDPC codes that are capacity-achieving for all rates R 1 R R 2 is shown. Based on the above observations, a method is proposed to design good punctured LDPC codes over a broad range of rates. Finally, it is shown that the results of this paper may be used for the proof of the existence of the capacity-achieving LDPC codes over binary-input-outputsymmetric memoryless channels.

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“…One can then proceed as in [36] and show that the AWGNC, the BSC, and the BEC pseudo-weights [3], [22], [23], [54], [55], [58] ofω will be at least as large as the corresponding pseudo-weights of ω. As a corollary, the minimum AWGNC, BSC, and BEC pseudo-weights ofH are, respectively, at least as large as the corresponding minimum pseudo-weights of H. Similar results can also be obtained for the minimum Hamming distance.…”

Section: Lemma 24 Let the Parity-check Matricesh And H Be Such That Tmentioning

confidence: 99%

Deriving Good LDPC Convolutional Codes from LDPC Block Codes

Pusane

Smarandache

Vontobel

et al. 2011

IEEE Trans. Inform. Theory

119

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Abstract-Low-density parity-check (LDPC) convolutional codes are capable of achieving excellent performance with low encoding and decoding complexity. In this paper we discuss several graph-cover-based methods for deriving families of timeinvariant and time-varying LDPC convolutional codes from LDPC block codes and show how earlier proposed LDPC convolutional code constructions can be presented within this framework.Some of the constructed convolutional codes significantly outperform the underlying LDPC block codes. We investigate some possible reasons for this "convolutional gain," and we also discuss the -mostly moderate -decoder cost increase that is incurred by going from LDPC block to LDPC convolutional codes.

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On the Effective Weights of Pseudocodewords for Codes Defined on Graphs with Cycles

Cited by 96 publications

References 8 publications

Graphical Models, Exponential Families, and Variational Inference

Graphical Models, Exponential Families, and Variational Inference

Results on Punctured Low-Density Parity-Check Codes and Improved Iterative Decoding Techniques

Deriving Good LDPC Convolutional Codes from LDPC Block Codes

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