Local Optimality Certificates for LP Decoding of Tanner Codes

We present a new combinatorial characterization for local optimality of a codeword in an irregular Tanner code. The main novelty in this characterization is that it is based on a linear combination of subtrees in the computation trees. These subtrees may have any degree in the local code nodes and may have any height (even greater than the girth). We expect this new characterization to lead to improvements in bounds for successful decoding. We prove that local optimality in this new characterization implies ML-optimality and LP-optimality, as one would expect. Finally, we show that is possible to compute efficiently a certificate for the local optimality of a codeword given an LLR vector

Statistical physics methods for sparse graph codes

This thesis deals with the asymptotic analysis of coding systems based on sparse graph codes. The goal of this work is to analyze the decoder performance when transmitting over a general binary-input memoryless symmetric-output (BMS) channel. We consider the two most fundamental decoders, the optimal maximum a posteriori (MAP) decoder and the sub-optimal belief propagation (BP) decoder. The BP decoder has low-complexity and its performance analysis is, hence, of great interest. The MAP decoder, on the other hand, is computationally expensive. However, the MAP decoder analysis provides fundamental limits on the code performance. As a result, the MAP-decoding analysis is important in designing codes which achieve the ultimate Shannon limit. It would be fair to say that, over the binary erasure channel (BEC), the performance of the MAP and BP decoder has been thoroughly understood. However, much less is known in the case of transmission over general BMS channels. The combinatorial methods used for analyzing the case of BEC do not extend easily to the general case. The main goal of this thesis is to advance the analysis in the case of transmission over general BMS channels. To do this, we use the recent convergence of statistical physics and coding theory. Sparse graph codes can be mapped into appropriate statistical physics spin-glass models. This allows us to use sophisticated methods from rigorous statistical mechanics like the correlation inequalities, interpolation method and cluster expansions for the purpose of our analysis. One of the main results of this thesis is that in some regimes of noise, the BP decoder is optimal for a typical code in an ensemble of codes. This result is a pleasing extension of the same result for the case of BEC. An important consequence of our results is that the heuristic predictions of the replica and cavity methods of spin-glass theory are correct in the realm of sparse graph codes

Community detection and stochastic block models: recent developments

The stochastic block model (SBM) is a random graph model with planted clusters. It is widely employed as a canonical model to study clustering and community detection, and provides generally a fertile ground to study the statistical and computational tradeoffs that arise in network and data sciences. This note surveys the recent developments that establish the fundamental limits for community detection in the SBM, both with respect to information-theoretic and computational thresholds, and for various recovery requirements such as exact, partial and weak recovery (a.k.a., detection). The main results discussed are the phase transitions for exact recovery at the Chernoff-Hellinger threshold, the phase transition for weak recovery at the Kesten-Stigum threshold, the optimal distortion-SNR tradeoff for partial recovery, the learning of the SBM parameters and the gap between information-theoretic and computational thresholds. The note also covers some of the algorithms developed in the quest of achieving the limits, in particular two-round algorithms via graph-splitting, semi-definite programming, linearized belief propagation, classical and nonbacktracking spectral methods. A few open problems are also discussed

Convex relaxation methods for graphical models : Lagrangian and maximum entropy approaches

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 241-257).Graphical models provide compact representations of complex probability distributions of many random variables through a collection of potential functions defined on small subsets of these variables. This representation is defined with respect to a graph in which nodes represent random variables and edges represent the interactions among those random variables. Graphical models provide a powerful and flexible approach to many problems in science and engineering, but also present serious challenges owing to the intractability of optimal inference and estimation over general graphs. In this thesis, we consider convex optimization methods to address two central problems that commonly arise for graphical models. First, we consider the problem of determining the most probable configuration-also known as the maximum a posteriori (MAP) estimate-of all variables in a graphical model, conditioned on (possibly noisy) measurements of some variables. This general problem is intractable, so we consider a Lagrangian relaxation (LR) approach to obtain a tractable dual problem. This involves using the Lagrangian decomposition technique to break up an intractable graph into tractable subgraphs, such as small "blocks" of nodes, embedded trees or thin subgraphs. We develop a distributed, iterative algorithm that minimizes the Lagrangian dual function by block coordinate descent. This results in an iterative marginal-matching procedure that enforces consistency among the subgraphs using an adaptation of the well-known iterative scaling algorithm. This approach is developed both for discrete variable and Gaussian graphical models. In discrete models, we also introduce a deterministic annealing procedure, which introduces a temperature parameter to define a smoothed dual function and then gradually reduces the temperature to recover the (non-differentiable) Lagrangian dual. When strong duality holds, we recover the optimal MAP estimate. We show that this occurs for a broad class of "convex decomposable" Gaussian graphical models, which generalizes the "pairwise normalizable" condition known to be important for iterative estimation in Gaussian models.(cont.) In certain "frustrated" discrete models a duality gap can occur using simple versions of our approach. We consider methods that adaptively enhance the dual formulation, by including more complex subgraphs, so as to reduce the duality gap. In many cases we are able to eliminate the duality gap and obtain the optimal MAP estimate in a tractable manner. We also propose a heuristic method to obtain approximate solutions in cases where there is a duality gap. Second, we consider the problem of learning a graphical model (both the graph and its potential functions) from sample data. We propose the maximum entropy relaxation (MER) method, which is the convex optimization problem of selecting the least informative (maximum entropy) model over an exponential family of graphical models subject to constraints that small subsets of variables should have marginal distributions that are close to the distribution of sample data. We use relative entropy to measure the divergence between marginal probability distributions. We find that MER leads naturally to selection of sparse graphical models. To identify this sparse graph efficiently, we use a "bootstrap" method that constructs the MER solution by solving a sequence of tractable subproblems defined over thin graphs, including new edges at each step to correct for large marginal divergences that violate the MER constraint. The MER problem on each of these subgraphs is efficiently solved using the primaldual interior point method (implemented so as to take advantage of efficient inference methods for thin graphical models). We also consider a dual formulation of MER that minimizes a convex function of the potentials of the graphical model. This MER dual problem can be interpreted as a robust version of maximum-likelihood parameter estimation, where the MER constraints specify the uncertainty in the sufficient statistics of the model. This also corresponds to a regularized maximum-likelihood approach, in which an information-geometric regularization term favors selection of sparse potential representations. We develop a relaxed version of the iterative scaling method to solve this MER dual problem.by Jason K. Johnson.Ph.D

LIPIcs, Volume 261, ICALP 2023, Complete Volume

LIPIcs, Volume 261, ICALP 2023, Complete Volum

Fast scrambles and event horizons

Tesis doctoral inédita. Universidad Autónoma de Madrid, Facultad de Ciencias, Departamento de Física Teórica. Fecha de lectura: 07-10-201

On The Analysis of Spatially-Coupled GLDPC Codes and The Weighted Min-Sum Algorithm

This dissertation studies methods to achieve reliable communication over unreliable channels. Iterative decoding algorithms for low-density parity-check (LDPC) codes and generalized LDPC (GLDPC) codes are analyzed. A new class of error-correcting codes to enhance the reliability of the communication for high-speed systems, such as optical communication systems, is proposed. The class of spatially-coupled GLDPC codes is studied, and a new iterative hard- decision decoding (HDD) algorithm for GLDPC codes is introduced. The main result is that the minimal redundancy allowed by Shannon’s Channel Coding Theorem can be achieved by using the new iterative HDD algorithm with spatially-coupled GLDPC codes. A variety of low-density parity-check (LDPC) ensembles have now been observed to approach capacity with iterative decoding. However, all of them use soft (i.e., non-binary) messages and a posteriori probability (APP) decoding of their component codes. To the best of our knowledge, this is the first system that can approach the channel capacity using iterative HDD. The optimality of a codeword returned by the weighted min-sum (WMS) algorithm, an iterative decoding algorithm which is widely used in practice, is studied as well. The attenuated max-product (AttMP) decoding and weighted min-sum (WMS) decoding for LDPC codes are analyzed. Applying the max-product (and belief- propagation) algorithms to loopy graphs are now quite popular for best assignment problems. This is largely due to their low computational complexity and impressive performance in practice. Still, there is no general understanding of the conditions required for convergence and/or the optimality of converged solutions. This work presents an analysis of both AttMP decoding and WMS decoding for LDPC codes which guarantees convergence to a fixed point when a weight factor, β, is sufficiently small. It also shows that, if the fixed point satisfies some consistency conditions, then it must be both a linear-programming (LP) and maximum-likelihood (ML) decoding solution