## 2015 |

Olmos, Pablo M; Mitchell, David G M; Costello, Daniel J Analyzing the Finite-Length Performance of Generalized LDPC Codes Inproceedings 2015 IEEE International Symposium on Information Theory (ISIT), pp. 2683–2687, IEEE, Hong Kong, 2015, ISBN: 978-1-4673-7704-1. Abstract | Links | BibTeX | Tags: BEC, binary codes, binary erasure channel, Block codes, Codes on graphs, Decoding, Differential equations, error probability, finite-length generalized LDPC block codes, finite-length performance analysis, generalized LDPC codes, generalized peeling decoder, GLDPC block codes, graph degree distribution, graph theory, Iterative decoding, parity check codes, protographs @inproceedings{Olmos2015b, title = {Analyzing the Finite-Length Performance of Generalized LDPC Codes}, author = {Pablo M Olmos and David G M Mitchell and Daniel J Costello}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7282943}, doi = {10.1109/ISIT.2015.7282943}, isbn = {978-1-4673-7704-1}, year = {2015}, date = {2015-06-01}, booktitle = {2015 IEEE International Symposium on Information Theory (ISIT)}, pages = {2683--2687}, publisher = {IEEE}, address = {Hong Kong}, abstract = {In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved.}, keywords = {BEC, binary codes, binary erasure channel, Block codes, Codes on graphs, Decoding, Differential equations, error probability, finite-length generalized LDPC block codes, finite-length performance analysis, generalized LDPC codes, generalized peeling decoder, GLDPC block codes, graph degree distribution, graph theory, Iterative decoding, parity check codes, protographs}, pubstate = {published}, tppubtype = {inproceedings} } In this paper, we analyze the performance of finite-length generalized LDPC (GLDPC) block codes constructed from protographs when transmission takes place over the binary erasure channel (BEC). A generalized peeling decoder is proposed and we derive a system of differential equations that gives the expected evolution of the graph degree distribution during decoding. We then show that the finite-length performance of a GLDPC code can be estimated by means of a simple scaling law, where a single scaling parameter represents the finite-length properties of the code. We also show that, as we consider stronger component codes, both the asymptotic threshold and the finite-length scaling parameter are improved. |

Olmos, Pablo M; Urbanke, Rudiger L A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes Journal Article IEEE Transactions on Information Theory, 61 (6), pp. 3164–3184, 2015, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, Journal, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015bb, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Pablo M Olmos and Rudiger L Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7086074}, doi = {10.1109/TIT.2015.2422816}, issn = {0018-9448}, year = {2015}, date = {2015-06-01}, journal = {IEEE Transactions on Information Theory}, volume = {61}, number = {6}, pages = {3164--3184}, abstract = {Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.}, keywords = {asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, Journal, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes}, pubstate = {published}, tppubtype = {article} } Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints. |

Olmos, Pablo M; Urbanke, Rudiger L A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes Journal Article IEEE Transactions on Information Theory, 61 (6), pp. 3164–3184, 2015, ISSN: 0018-9448. Abstract | Links | BibTeX | Tags: asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes @article{Olmos2015c, title = {A Scaling Law to Predict the Finite-Length Performance of Spatially-Coupled LDPC Codes}, author = {Pablo M Olmos and Rudiger L Urbanke}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=7086074}, doi = {10.1109/TIT.2015.2422816}, issn = {0018-9448}, year = {2015}, date = {2015-06-01}, journal = {IEEE Transactions on Information Theory}, volume = {61}, number = {6}, pages = {3164--3184}, abstract = {Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints.}, keywords = {asymptotic analysis, asymptotic properties, binary erasure channel, Channel Coding, Codes on graphs, Couplings, Decoding, Differential equations, error probability, finite length performance, finite length spatially coupled code, finite-length code performance, finite-length performance, Iterative decoding, iterative decoding thresholds, parity check codes, Probability, SC-LDPC codes, scaling law, Sockets, spatially coupled LDPC codes, spatially-coupled LDPC codes}, pubstate = {published}, tppubtype = {article} } Spatially-coupled low-density parity-check (SC-LDPC) codes are known to have excellent asymptotic properties. Much less is known regarding their finite-length performance. We propose a scaling law to predict the error probability of finite-length spatially coupled code ensembles when transmission takes place over the binary erasure channel. We discuss how the parameters of the scaling law are connected to fundamental quantities appearing in the asymptotic analysis of these ensembles and we verify that the predictions of the scaling law fit well to the data derived from simulations over a wide range of parameters. The ultimate goal of this line of research is to develop analytic tools for the design of SC-LDPC codes under practical constraints. |

## 2013 |

Alvarez, Mauricio; Luengo, David; Lawrence, Neil D Linear Latent Force Models Using Gaussian Processes Journal Article IEEE Trans. Pattern Anal. Mach. Intell., 35 (11), pp. 2693–2705, 2013. Abstract | Links | BibTeX | Tags: Analytical models, Computational modeling, Data models, Differential equations, Force, Gaussian processes, Mathematical mode @article{Alvarez2013, title = {Linear Latent Force Models Using Gaussian Processes}, author = {Mauricio Alvarez and David Luengo and Neil D Lawrence}, url = {http://dblp.uni-trier.de/db/journals/pami/pami35.html#AlvarezLL13 http://ieeexplore.ieee.org/xpl/articleDetails.jsp?arnumber=6514873}, year = {2013}, date = {2013-01-01}, journal = {IEEE Trans. Pattern Anal. Mach. Intell.}, volume = {35}, number = {11}, pages = {2693--2705}, abstract = {Purely data-driven approaches for machine learning present difficulties when data are scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data-driven modeling with a physical model of the system. We show how different, physically inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology, and geostatistics.}, keywords = {Analytical models, Computational modeling, Data models, Differential equations, Force, Gaussian processes, Mathematical mode}, pubstate = {published}, tppubtype = {article} } Purely data-driven approaches for machine learning present difficulties when data are scarce relative to the complexity of the model or when the model is forced to extrapolate. On the other hand, purely mechanistic approaches need to identify and specify all the interactions in the problem at hand (which may not be feasible) and still leave the issue of how to parameterize the system. In this paper, we present a hybrid approach using Gaussian processes and differential equations to combine data-driven modeling with a physical model of the system. We show how different, physically inspired, kernel functions can be developed through sensible, simple, mechanistic assumptions about the underlying system. The versatility of our approach is illustrated with three case studies from motion capture, computational biology, and geostatistics. |

## 2011 |

Olmos, Pablo M; Murillo-Fuentes, Juan Jose; Perez-Cruz, Fernando Capacity Achieving LDPC Ensembles for the TEP Decoder in Erasure Channels Inproceedings 2011 IEEE International Symposium on Information Theory Proceedings, pp. 2398–2402, IEEE, St. Petersburg, 2011, ISSN: 2157-8095. Abstract | Links | BibTeX | Tags: BP threshold, Complexity theory, Decoding, Differential equations, erasure channels, fixed-rate code, Iterative decoding, LDPC, low-density parity-check codes, MAP capacity, MAP threshold, optimisation, Optimization, optimization problem, parity check codes, TEP decoder, tree-expectation propagation decoder @inproceedings{Olmos2011b, title = {Capacity Achieving LDPC Ensembles for the TEP Decoder in Erasure Channels}, author = {Pablo M Olmos and Juan Jose Murillo-Fuentes and Fernando Perez-Cruz}, url = {http://ieeexplore.ieee.org/lpdocs/epic03/wrapper.htm?arnumber=6033993}, issn = {2157-8095}, year = {2011}, date = {2011-01-01}, booktitle = {2011 IEEE International Symposium on Information Theory Proceedings}, pages = {2398--2402}, publisher = {IEEE}, address = {St. Petersburg}, abstract = {In this work we address the design of degree distributions (DD) of low-density parity-check (LDPC) codes for the tree-expectation propagation (TEP) decoder. The optimization problem to find distributions to maximize the TEP decoding threshold for a fixed-rate code can not be analytically solved. We derive a simplified optimization problem that can be easily solved since it is based in the analytic expressions of the peeling decoder. Two kinds of solutions are obtained from this problem: we either design LDPC ensembles for which the BP threshold equals the MAP threshold or we get LDPC ensembles for which the TEP threshold outperforms the BP threshold, even achieving the MAP capacity in some cases. Hence, we proved that there exist ensembles for which the MAP solution can be obtained with linear complexity even though the BP threshold does not achieve the MAP threshold.}, keywords = {BP threshold, Complexity theory, Decoding, Differential equations, erasure channels, fixed-rate code, Iterative decoding, LDPC, low-density parity-check codes, MAP capacity, MAP threshold, optimisation, Optimization, optimization problem, parity check codes, TEP decoder, tree-expectation propagation decoder}, pubstate = {published}, tppubtype = {inproceedings} } In this work we address the design of degree distributions (DD) of low-density parity-check (LDPC) codes for the tree-expectation propagation (TEP) decoder. The optimization problem to find distributions to maximize the TEP decoding threshold for a fixed-rate code can not be analytically solved. We derive a simplified optimization problem that can be easily solved since it is based in the analytic expressions of the peeling decoder. Two kinds of solutions are obtained from this problem: we either design LDPC ensembles for which the BP threshold equals the MAP threshold or we get LDPC ensembles for which the TEP threshold outperforms the BP threshold, even achieving the MAP capacity in some cases. Hence, we proved that there exist ensembles for which the MAP solution can be obtained with linear complexity even though the BP threshold does not achieve the MAP threshold. |