arXiv subjects

Igal Sason

Publications and source records attributed to Igal Sason.

At least 37 records · Page 2Linked to original sources

arXiv2020

On Relations Between the Relative entropy and $χ^2$-Divergence, Generalizations and Applications

The relative entropy and chi-squared divergence are fundamental divergence measures in information theory and statistics. This paper is focused on a study of integral relations between the two divergences, the implications of these relations, their information-theoretic applications, and some generalizations pertaining to the rich class of $f$-divergences. Applications that are studied in this paper refer to lossless compression, the method of types and large deviations, strong~data-processing inequalities, bounds on contraction coefficients and maximal correlation, and the convergence rate to stationarity of a type of discrete-time Markov chains.

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arXiv2020

An Integral Representation of the Logarithmic Function with Applications in Information Theory

We explore a well-known integral representation of the logarithmic function, and demonstrate its usefulness in obtaining compact, easily-computable exact formulas for quantities that involve expectations and higher moments of the logarithm of a positive random variable (or the logarithm of a sum of positive random variables). The integral representation of the logarithm is proved useful in a variety of information-theoretic applications, including universal lossless data compression, entropy and differential entropy evaluations, and the calculation of the ergodic capacity of the single-input, multiple-output (SIMO) Gaussian channel with random parameters (known to both transmitter and receiver). This integral representation and its variants are anticipated to serve as a useful tool in additional applications, as a rigorous alternative to the popular (but non-rigorous) replica method (at least in some situations).

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arXiv2019

On Data-Processing and Majorization Inequalities for $f$-Divergences with Applications

This paper is focused on derivations of data-processing and majorization inequalities for $f$-divergences, and their applications in information theory and statistics. For the accessibility of the material, the main results are first introduced without proofs, followed by exemplifications of the theorems with further related analytical results, interpretations, and information-theoretic applications. One application refers to the performance analysis of list decoding with either fixed or variable list sizes; some earlier bounds on the list decoding error probability are reproduced in a unified way, and new bounds are obtained and exemplified numerically. Another application is related to a study of the quality of approximating a probability mass function, induced by the leaves of a Tunstall tree, by an equiprobable distribution. The compression rates of finite-length Tunstall codes are further analyzed for asserting their closeness to the Shannon entropy of a memoryless and stationary discrete source. Almost all the analysis is relegated to the appendices, which form a major part of this manuscript.

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arXiv2018

Tight Bounds on the Rényi Entropy via Majorization with Applications to Guessing and Compression

This paper provides tight bounds on the Rényi entropy of a function of a discrete random variable with a finite number of possible values, where the considered function is not one-to-one. To that end, a tight lower bound on the Rényi entropy of a discrete random variable with a finite support is derived as a function of the size of the support, and the ratio of the maximal to minimal probability masses. This work was inspired by the recently published paper by Cicalese et al., which is focused on the Shannon entropy, and it strengthens and generalizes the results of that paper to Rényi entropies of arbitrary positive orders. In view of these generalized bounds and the works by Arikan and Campbell, non-asymptotic bounds are derived for guessing moments and lossless data compression of discrete memoryless sources.

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arXiv2018

Improved Bounds on Lossless Source Coding and Guessing Moments via Rényi Measures

This paper provides upper and lower bounds on the optimal guessing moments of a random variable taking values on a finite set when side information may be available. These moments quantify the number of guesses required for correctly identifying the unknown object and, similarly to Arikan's bounds, they are expressed in terms of the Arimoto-Rényi conditional entropy. Although Arikan's bounds are asymptotically tight, the improvement of the bounds in this paper is significant in the non-asymptotic regime. Relationships between moments of the optimal guessing function and the MAP error probability are also established, characterizing the exact locus of their attainable values. The bounds on optimal guessing moments serve to improve non-asymptotic bounds on the cumulant generating function of the codeword lengths for fixed-to-variable optimal lossless source coding without prefix constraints. Non-asymptotic bounds on the reliability function of discrete memoryless sources are derived as well. Relying on these techniques, lower bounds on the cumulant generating function of the codeword lengths are derived, by means of the smooth Rényi entropy, for source codes that allow decoding errors.

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arXiv2018

On $f$-Divergences: Integral Representations, Local Behavior, and Inequalities

This paper is focused on $f$-divergences, consisting of three main contributions. The first one introduces integral representations of a general $f$-divergence by means of the relative information spectrum. The second part provides a new approach for the derivation of $f$-divergence inequalities, and it exemplifies their utility in the setup of Bayesian binary hypothesis testing. The last part of this paper further studies the local behavior of $f$-divergences.

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arXiv2017

Arimoto-Rényi Conditional Entropy and Bayesian $M$-ary Hypothesis Testing

This paper gives upper and lower bounds on the minimum error probability of Bayesian $M$-ary hypothesis testing in terms of the Arimoto-Rényi conditional entropy of an arbitrary order $α$. The improved tightness of these bounds over their specialized versions with the Shannon conditional entropy ($α=1$) is demonstrated. In particular, in the case where $M$ is finite, we show how to generalize Fano's inequality under both the conventional and list-decision settings. As a counterpart to the generalized Fano's inequality, allowing $M$ to be infinite, a lower bound on the Arimoto-Rényi conditional entropy is derived as a function of the minimum error probability. Explicit upper and lower bounds on the minimum error probability are obtained as a function of the Arimoto-Rényi conditional entropy for both positive and negative $α$. Furthermore, we give upper bounds on the minimum error probability as functions of the Rényi divergence. In the setup of discrete memoryless channels, we analyze the exponentially vanishing decay of the Arimoto-Rényi conditional entropy of the transmitted codeword given the channel output when averaged over a random coding ensemble.

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arXiv2016

$f$-divergence Inequalities

This paper develops systematic approaches to obtain $f$-divergence inequalities, dealing with pairs of probability measures defined on arbitrary alphabets. Functional domination is one such approach, where special emphasis is placed on finding the best possible constant upper bounding a ratio of $f$-divergences. Another approach used for the derivation of bounds among $f$-divergences relies on moment inequalities and the logarithmic-convexity property, which results in tight bounds on the relative entropy and Bhattacharyya distance in terms of $χ^2$ divergences. A rich variety of bounds are shown to hold under boundedness assumptions on the relative information. Special attention is devoted to the total variation distance and its relation to the relative information and relative entropy, including "reverse Pinsker inequalities," as well as on the $E_γ$ divergence, which generalizes the total variation distance. Pinsker's inequality is extended for this type of $f$-divergence, a result which leads to an inequality linking the relative entropy and relative information spectrum. Integral expressions of the Rényi divergence in terms of the relative information spectrum are derived, leading to bounds on the Rényi divergence in terms of either the variational distance or relative entropy.

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arXiv2016

$f$-Divergence Inequalities via Functional Domination

This paper considers derivation of $f$-divergence inequalities via the approach of functional domination. Bounds on an $f$-divergence based on one or several other $f$-divergences are introduced, dealing with pairs of probability measures defined on arbitrary alphabets. In addition, a variety of bounds are shown to hold under boundedness assumptions on the relative information. The journal paper, which includes more approaches for the derivation of f-divergence inequalities and proofs, is available on the arXiv at https://arxiv.org/abs/1508.00335, and it has been published in the IEEE Trans. on Information Theory, vol. 62, no. 11, pp. 5973-6006, November 2016.

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arXiv2016

Projection Theorems for the Rényi Divergence on $α$-Convex Sets

This paper studies forward and reverse projections for the Rényi divergence of order $α\in (0, \infty)$ on $α$-convex sets. The forward projection on such a set is motivated by some works of Tsallis {\em et al.} in statistical physics, and the reverse projection is motivated by robust statistics. In a recent work, van Erven and Harremoës proved a Pythagorean inequality for Rényi divergences on $α$-convex sets under the assumption that the forward projection exists. Continuing this study, a sufficient condition for the existence of forward projection is proved for probability measures on a general alphabet. For $α\in (1, \infty)$, the proof relies on a new Apollonius theorem for the Hellinger divergence, and for $α\in (0,1)$, the proof relies on the Banach-Alaoglu theorem from functional analysis. Further projection results are then obtained in the finite alphabet setting. These include a projection theorem on a specific $α$-convex set, which is termed an {\em $α$-linear family}, generalizing a result by Csiszár for $α\neq 1$. The solution to this problem yields a parametric family of probability measures which turns out to be an extension of the exponential family, and it is termed an {\em $α$-exponential family}. An orthogonality relationship between the $α$-exponential and $α$-linear families is established, and it is used to turn the reverse projection on an $α$-exponential family into a forward projection on a $α$-linear family. This paper also proves a convergence result of an iterative procedure used to calculate the forward projection on an intersection of a finite number of $α$-linear families.

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arXiv2016

On Renyi Entropy Power Inequalities

This paper gives improved Rényi entropy power inequalities (R-EPIs). Consider a sum $S_n = \sum_{k=1}^n X_k$ of $n$ independent continuous random vectors taking values on $\mathbb{R}^d$, and let $α\in [1, \infty]$. An R-EPI provides a lower bound on the order-$α$ Rényi entropy power of $S_n$ that, up to a multiplicative constant (which may depend in general on $n, α, d$), is equal to the sum of the order-$α$ Rényi entropy powers of the $n$ random vectors $\{X_k\}_{k=1}^n$. For $α=1$, the R-EPI coincides with the well-known entropy power inequality by Shannon. The first improved R-EPI is obtained by tightening the recent R-EPI by Bobkov and Chistyakov which relies on the sharpened Young's inequality. A further improvement of the R-EPI also relies on convex optimization and results on rank-one modification of a real-valued diagonal matrix.

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arXiv2015

On the Renyi Divergence, Joint Range of Relative Entropies, and a Channel Coding Theorem

This paper starts by considering the minimization of the Renyi divergence subject to a constraint on the total variation distance. Based on the solution of this optimization problem, the exact locus of the points $\bigl( D(Q\|P_1), D(Q\|P_2) \bigr)$ is determined when $P_1, P_2, Q$ are arbitrary probability measures which are mutually absolutely continuous, and the total variation distance between $P_1$ and $P_2$ is not below a given value. It is further shown that all the points of this convex region are attained by probability measures which are defined on a binary alphabet. This characterization yields a geometric interpretation of the minimal Chernoff information subject to a constraint on the total variation distance. This paper also derives an exponential upper bound on the performance of binary linear block codes (or code ensembles) under maximum-likelihood decoding. Its derivation relies on the Gallager bounding technique, and it reproduces the Shulman-Feder bound as a special case. The bound is expressed in terms of the Renyi divergence from the normalized distance spectrum of the code (or the average distance spectrum of the ensemble) to the binomially distributed distance spectrum of the capacity-achieving ensemble of random block codes. This exponential bound provides a quantitative measure of the degradation in performance of binary linear block codes (or code ensembles) as a function of the deviation of their distance spectra from the binomial distribution. An efficient use of this bound is considered.

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arXiv2015

Upper Bounds on the Relative Entropy and Rényi Divergence as a Function of Total Variation Distance for Finite Alphabets

A new upper bound on the relative entropy is derived as a function of the total variation distance for probability measures defined on a common finite alphabet. The bound improves a previously reported bound by Csiszár and Talata. It is further extended to an upper bound on the Rényi divergence of an arbitrary non-negative order (including $\infty$) as a function of the total variation distance.

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arXiv2015

Concentration of Measure Inequalities and Their Communication and Information-Theoretic Applications

During the last two decades, concentration of measure has been a subject of various exciting developments in convex geometry, functional analysis, statistical physics, high-dimensional statistics, probability theory, information theory, communications and coding theory, computer science, and learning theory. One common theme which emerges in these fields is probabilistic stability: complicated, nonlinear functions of a large number of independent or weakly dependent random variables often tend to concentrate sharply around their expected values. Information theory plays a key role in the derivation of concentration inequalities. Indeed, both the entropy method and the approach based on transportation-cost inequalities are two major information-theoretic paths toward proving concentration. This brief survey is based on a recent monograph of the authors in the Foundations and Trends in Communications and Information Theory (online available at http://arxiv.org/pdf/1212.4663v8.pdf), and a tutorial given by the authors at ISIT 2015. It introduces information theorists to three main techniques for deriving concentration inequalities: the martingale method, the entropy method, and the transportation-cost inequalities. Some applications in information theory, communications, and coding theory are used to illustrate the main ideas.

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arXiv2015

On Reverse Pinsker Inequalities

New upper bounds on the relative entropy are derived as a function of the total variation distance. One bound refines an inequality by Verdú for general probability measures. A second bound improves the tightness of an inequality by Csiszár and Talata for arbitrary probability measures that are defined on a common finite set. The latter result is further extended, for probability measures on a finite set, leading to an upper bound on the Rényi divergence of an arbitrary non-negative order (including $\infty$) as a function of the total variation distance. Another lower bound by Verdú on the total variation distance, expressed in terms of the distribution of the relative information, is tightened and it is attained under some conditions. The effect of these improvements is exemplified.

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arXiv2015

On the Corner Points of the Capacity Region of a Two-User Gaussian Interference Channel

This work considers the corner points of the capacity region of a two-user Gaussian interference channel (GIC). In a two-user GIC, the rate pairs where one user transmits its data at the single-user capacity (without interference), and the other at the largest rate for which reliable communication is still possible are called corner points. This paper relies on existing outer bounds on the capacity region of a two-user GIC that are used to derive informative bounds on the corner points of the capacity region. The new bounds refer to a weak two-user GIC (i.e., when both cross-link gains in standard form are positive and below 1), and a refinement of these bounds is obtained for the case where the transmission rate of one user is within $\varepsilon > 0$ of the single-user capacity. The bounds on the corner points are asymptotically tight as the transmitted powers tend to infinity, and they are also useful for the case of moderate SNR and INR. Upper and lower bounds on the gap (denoted by $Δ$) between the sum-rate and the maximal achievable total rate at the two corner points are derived. This is followed by an asymptotic analysis analogous to the study of the generalized degrees of freedom (where the SNR and INR scalings are coupled such that $\frac{\log(\text{INR})}{\log(\text{SNR})} = α\geq 0$), leading to an asymptotic characterization of this gap which is exact for the whole range of $α$. The upper and lower bounds on $Δ$ are asymptotically tight in the sense that they achieve the exact asymptotic characterization. Improved bounds on $Δ$ are derived for finite SNR and INR, and their improved tightness is exemplified numerically.

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arXiv2015

Tight Bounds for Symmetric Divergence Measures and a Refined Bound for Lossless Source Coding

Tight bounds for several symmetric divergence measures are derived in terms of the total variation distance. It is shown that each of these bounds is attained by a pair of 2 or 3-element probability distributions. An application of these bounds for lossless source coding is provided, refining and improving a certain bound by Csiszár. Another application of these bounds has been recently introduced by Yardi. et al. for channel-code detection.

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arXiv2015

Tight Bounds for Symmetric Divergence Measures and a New Inequality Relating $f$-Divergences

Tight bounds for several symmetric divergence measures are introduced, given in terms of the total variation distance. Each of these bounds is attained by a pair of 2 or 3-element probability distributions. An application of these bounds for lossless source coding is provided, refining and improving a certain bound by Csiszár. A new inequality relating $f$-divergences is derived, and its use is exemplified. The last section of this conference paper is not included in the recent journal paper that was published in the February 2015 issue of the IEEE Trans. on Information Theory (see arXiv:1403.7164), as well as some new paragraphs throughout the paper which are linked to new references.

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