Efficient sphere-covering and converse measure concentration via generalized coding theorems

Suppose A is a finite set equipped with a probability measure P and let M be a ``mass'' function on A. We give a probabilistic characterization of the most efficient way in which A^n can be almost-covered using spheres of a fixed radius. An almost-covering is a subset C_n of A^n, such that the union of the spheres centered at the points of C_n has probability close to one with respect to the product measure P^n. An efficient covering is one with small mass M^n(C_n); n is typically large. With different choices for M and the geometry on A our results give various corollaries as special cases, including Shannon's data compression theorem, a version of Stein's lemma (in hypothesis testing), and a new converse to some measure concentration inequalities on discrete spaces. Under mild conditions, we generalize our results to abstract spaces and non-product measures.Comment: 29 pages. See also http://www.stat.purdue.edu/~yiannis

Asymptotic Estimates in Information Theory with Non-Vanishing Error Probabilities

This monograph presents a unified treatment of single- and multi-user problems in Shannon's information theory where we depart from the requirement that the error probability decays asymptotically in the blocklength. Instead, the error probabilities for various problems are bounded above by a non-vanishing constant and the spotlight is shone on achievable coding rates as functions of the growing blocklengths. This represents the study of asymptotic estimates with non-vanishing error probabilities. In Part I, after reviewing the fundamentals of information theory, we discuss Strassen's seminal result for binary hypothesis testing where the type-I error probability is non-vanishing and the rate of decay of the type-II error probability with growing number of independent observations is characterized. In Part II, we use this basic hypothesis testing result to develop second- and sometimes, even third-order asymptotic expansions for point-to-point communication. Finally in Part III, we consider network information theory problems for which the second-order asymptotics are known. These problems include some classes of channels with random state, the multiple-encoder distributed lossless source coding (Slepian-Wolf) problem and special cases of the Gaussian interference and multiple-access channels. Finally, we discuss avenues for further research.Comment: Further comments welcom

Data compression with low distortion and finite blocklength

This paper considers lossy source coding of n-dimensional continuous memoryless sources with low mean-square error distortion and shows a simple, explicit approximation to the minimum source coding rate. More precisely, a nonasymptotic version of Shannon's lower bound is presented. Lattice quantizers are shown to approach that lower bound, provided that the source density is smooth enough and the distortion is low, which implies that fine multidimensional lattice coverings are nearly optimal in the rate-distortion sense even at finite n. The achievability proof technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal

Concentration of Measure Inequalities in Information Theory, Communications and Coding (Second Edition)

During the last two decades, concentration inequalities have been the subject of exciting developments in various areas, including convex geometry, functional analysis, statistical physics, high-dimensional statistics, pure and applied probability theory, information theory, theoretical computer science, and learning theory. This monograph focuses on some of the key modern mathematical tools that are used for the derivation of concentration inequalities, on their links to information theory, and on their various applications to communications and coding. In addition to being a survey, this monograph also includes various new recent results derived by the authors. The first part of the monograph introduces classical concentration inequalities for martingales, as well as some recent refinements and extensions. The power and versatility of the martingale approach is exemplified in the context of codes defined on graphs and iterative decoding algorithms, as well as codes for wireless communication. The second part of the monograph introduces the entropy method, an information-theoretic technique for deriving concentration inequalities. The basic ingredients of the entropy method are discussed first in the context of logarithmic Sobolev inequalities, which underlie the so-called functional approach to concentration of measure, and then from a complementary information-theoretic viewpoint based on transportation-cost inequalities and probability in metric spaces. Some representative results on concentration for dependent random variables are briefly summarized, with emphasis on their connections to the entropy method. Finally, we discuss several applications of the entropy method to problems in communications and coding, including strong converses, empirical distributions of good channel codes, and an information-theoretic converse for concentration of measure.Comment: Foundations and Trends in Communications and Information Theory, vol. 10, no 1-2, pp. 1-248, 2013. Second edition was published in October 2014. ISBN to printed book: 978-1-60198-906-

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

Data Compression with Low Distortion and Finite Blocklength

This paper considers lossy source coding of n-dimensional memoryless sources and shows an explicit approximation to the minimum source coding rate required to sustain the probability of exceeding distortion d no greater than ϵ, which is simpler than known dispersion-based approximations. Our approach takes inspiration in the celebrated classical result stating that the Shannon lower bound to rate-distortion function becomes tight in the limit d → 0. We formulate an abstract version of the Shannon lower bound that recovers both the classical Shannon lower bound and the rate-distortion function itself as special cases. Likewise, we show that a nonasymptotic version of the abstract Shannon lower bound recovers all previously known nonasymptotic converses. A necessary and sufficient condition for the Shannon lower bound to be attained exactly is presented. It is demonstrated that whenever that condition is met, the rate-dispersion function is given simply by the varentropy of the source. Remarkably, all finite alphabet sources with balanced distortion measures satisfy that condition in the range of low distortions. Most continuous sources violate that condition. Still, we show that lattice quantizers closely approach the nonasymptotic Shannon lower bound, provided that the source density is smooth enough and the distortion is low. This implies that fine multidimensional lattice coverings are nearly optimal in the rate-distortion sense even at finite . The achievability proof technique is based on a new bound on the output entropy of lattice quantizers in terms of the differential entropy of the source, the lattice cell size, and a smoothness parameter of the source density. The technique avoids both the usual random coding argument and the simplifying assumption of the presence of a dither signal

On the Reliability Function of Distributed Hypothesis Testing Under Optimal Detection

The distributed hypothesis testing problem with full side-information is studied. The trade-off (reliability function) between the two types of error exponents under limited rate is studied in the following way. First, the problem is reduced to the problem of determining the reliability function of channel codes designed for detection (in analogy to a similar result which connects the reliability function of distributed lossless compression and ordinary channel codes). Second, a single-letter random-coding bound based on a hierarchical ensemble, as well as a single-letter expurgated bound, are derived for the reliability of channel-detection codes. Both bounds are derived for a system which employs the optimal detection rule. We conjecture that the resulting random-coding bound is ensemble-tight, and consequently optimal within the class of quantization-and-binning schemes

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization