A Counter-Example to the Mismatched Decoding Converse for Binary-Input Discrete Memoryless Channels

This paper studies the mismatched decoding problem for binary-input discrete memoryless channels. An example is provided for which an achievable rate based on superposition coding exceeds the LM rate (Hui, 1983; Csisz\'ar-K\"orner, 1981), thus providing a counter-example to a previously reported converse result (Balakirsky, 1995). Both numerical evaluations and theoretical results are used in establishing this claim.Comment: Extended version of paper accepted to IEEE Transactions on Information Theory; rate derivation and numerical algorithms included in appendice

Information-Theoretic Foundations of Mismatched Decoding

Shannon's channel coding theorem characterizes the maximal rate of information that can be reliably transmitted over a communication channel when optimal encoding and decoding strategies are used. In many scenarios, however, practical considerations such as channel uncertainty and implementation constraints rule out the use of an optimal decoder. The mismatched decoding problem addresses such scenarios by considering the case that the decoder cannot be optimized, but is instead fixed as part of the problem statement. This problem is not only of direct interest in its own right, but also has close connections with other long-standing theoretical problems in information theory. In this monograph, we survey both classical literature and recent developments on the mismatched decoding problem, with an emphasis on achievable random-coding rates for memoryless channels. We present two widely-considered achievable rates known as the generalized mutual information (GMI) and the LM rate, and overview their derivations and properties. In addition, we survey several improved rates via multi-user coding techniques, as well as recent developments and challenges in establishing upper bounds on the mismatch capacity, and an analogous mismatched encoding problem in rate-distortion theory. Throughout the monograph, we highlight a variety of applications and connections with other prominent information theory problems.Comment: Published in Foundations and Trends in Communications and Information Theory (Volume 17, Issue 2-3

Information-Theoretic Foundations of Mismatched Decoding

Shannon’s channel coding theorem characterizes the maximal rate of information that can be reliably transmitted over a communication channel when optimal encoding and decoding strategies are used. In many scenarios, however, practical considerations such as channel uncertainty and implementation constraints rule out the use of an optimal decoder. The mismatched decoding problem addresses such scenarios by considering the case that the decoder cannot be optimized, but is instead fixed as part of the problem statement. This problem is not only of direct interest in its own right, but also has close connections with other long-standing theoretical problems in information theory. In this monograph, we survey both classical literature and recent developments on the mismatched decoding problem, with an emphasis on achievable random-coding rates for memoryless channels. We present two widely-considered achievable rates known as the generalized mutual information (GMI) and the LM rate, and overview their derivations and properties. In addition, we survey several improved rates via multi-user coding techniques, as well as recent developments and challenges in establishing upper bounds on the mismatch capacity, and an analogous mismatched encoding problem in rate-distortion theory. Throughout the monograph, we highlight a variety of applications and connections with other prominent information theory problems

A Single-Letter Upper Bound to the Mismatch Capacity

We derive a single-letter upper bound to the mismatched-decoding capacity for discrete memoryless channels. The bound is expressed as the mutual information of a transformation of the channel, such that a maximum-likelihood decoding error on the translated channel implies a mismatched-decoding error in the original channel. In particular, a strong converse is shown to hold for this upper-bound: if the rate exceeds the upper-bound, the probability of error tends to 1 exponentially when the block-length tends to infinity. We also show that the underlying optimization problem is a convex-concave problem and that an efficient iterative algorithm converges to the optimal solution. In addition, we show that, unlike achievable rates in the literature, the multiletter version of the bound does not improve. A number of examples are discussed throughout the paper.European Research Council under Grant 725411, and by the Spanish Ministry of Economy and Competitiveness under Grant TEC2016-78434-C3-1-R

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

A Sphere-Packing Error Exponent for Mismatched Decoding

We derive a sphere-packing error exponent for coded transmission over discrete memoryless channels with a fixed decoding metric. By studying the error probability of the code over an auxiliary channel, we find a lower bound to the probability of error of mismatched decoding. The bound is shown to decay exponentially for coding rates smaller than a new upper bound to the mismatch capacity. For rates higher than the new upper bound, the error probability is shown to be bounded away from zero. The new upper bound is shown to improve over previous upper bounds to the mismatch capacity

Multiuser Random Coding Techniques for Mismatched Decoding

This paper studies multiuser random coding techniques for channel coding with a given (possibly suboptimal) decoding rule. For the mismatched discrete memoryless multiple-access channel, an error exponent is obtained that is tight with respect to the ensemble average, and positive within the interior of Lapidoth's achievable rate region. This exponent proves the ensemble tightness of the exponent of Liu and Hughes in the case of maximum-likelihood decoding. An equivalent dual form of Lapidoth's achievable rate region is given, and the latter is shown to immediately extend to channels with infinite and continuous alphabets. In the setting of single-user mismatched decoding, similar analysis techniques are applied to a refined version of superposition coding, which is shown to achieve rates at least as high as standard superposition coding for any set of random-coding parameters

Local to global geometric methods in information theory

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2008.Includes bibliographical references (p. 201-203).This thesis treats several information theoretic problems with a unified geometric approach. The development of this approach was motivated by the challenges encountered while working on these problems, and in turn, the testing of the initial tools to these problems suggested numerous refinements and improvements on the geometric methods. In ergodic probabilistic settings, Sanov's theorem gives asymptotic estimates on the probabilities of very rare events. The theorem also characterizes the exponential decay of the probabilities, as the sample size grows, and the exponential rate is given by the minimization of a certain divergence expression. In his seminal paper, A Mathematical Theory of Communication, Shannon introduced two influential ideas to simplify the complex task of evaluating the performance of a coding scheme: the asymptotic perspective (in the number of channel uses) and the random coding argument. In this setting, Sanov's theorem can be used to analyze ergodic information theoretic problems, and the performance of a coding scheme can be estimated by expressions involving the divergence. One would then like to use a geometric intuition to solve these problems, but the divergence is not a distance and our naive geometric intuition may lead to incorrect conclusions. In information geometry, a specific differential geometric structure is introduced by means of "dual affine connections". The approach we take in this thesis is slightly different and is based on introducing additional asymptotic regimes to analyze the divergence expressions. The following two properties play an important role. The divergence may not be a distance, but locally (i.e., when its arguments are "close to each other"), the divergence behaves like a squared distance.(cont.) Moreover, globally (i.e., when its arguments have no local restriction), it also preserves certain properties satisfied by squared distances. Therefore, we develop the Very Noisy and Hermite transformations, as techniques to map our global information theoretic problems in local ones. Through this localization, our global divergence expressions reduce in the limit to expressions defined in an inner product space. This provides us with a valuable geometric insight to the global problems, as well as a strong tool to find counter-examples. Finally, in certain cases, we have been able to "lift" results proven locally to results proven globally.(cont.) Therefore, we develop the Very Noisy and Hermite transformations, as techniques to map our global information theoretic problems in local ones. Through this localization, our global divergence expressions reduce in the limit to expressions defined in an inner product space. This provides us with a valuable geometric insight to the global problems, as well as a strong tool to find counter-examples. Finally, in certain cases, we have been able to "lift" results proven locally to results proven globally. We consider the following three problems. First, we address the problem of finding good linear decoders (maximizing additive metrics) for compound discrete memoryless channels. Known universal decoders are not linear and most of them heavily depend on the finite alphabet assumption. We show that by using a finite number of additive metrics, we can construct decoders that are universal (capacity achieving) on most compound sets. We then consider additive Gaussian noise channels. For a given perturbation of a Gaussian input distribution, we define an operator that measures how much variation is induced in the output entropy. We found that the singular functions of this operator are the Hermite polynomials, and the singular values are the powers of a signal to noise ratio. We show, in particular, how to use this structure on a Gaussian interference channel to characterize a regime where interference should not be treated as noise. Finally, we consider multi-input multi-output channels and discuss the properties of the optimal input distributions, for various random fading matrix ensembles. In particular, we prove Telatar's conjecture on the covariance structure minimizing the outage probability for output dimension one and input dimensions less than one hundred.by Emmanuel Auguste Abbe.Ph.D