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A special issue of Entropy (ISSN 1099-4300). This special issue belongs to the section "Information Theory, Probability and Statistics".

Deadline for manuscript submissions: closed (15 November 2019) | Viewed by 33199

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Special Issue Editor

Prof. Dr. Amos Lapidoth

E-Mail Website
Guest Editor

Signal and Information Processing Laboratory, ETH Zurich, 8092 Zurich, Switzerland
Interests: multi-terminal information theory; the role of feedback in communication; digital communication; fading channels; optical communication
Special Issues, Collections and Topics in MDPI journals

Special Issue Information

Dear Colleagues,

Classical information measures such as entropy, relative entropy (Kullback–Leibler divergence), and mutual information have found numerous applications in storage, compression, transmission, cryptography, statistics, large deviations, gambling, and physics. However, over the years—arguably starting with the pioneering work of Alfréd Rényi (1921–1970)—other information measures were introduced and studied. Those include Rényi Entropy, Rényi Divergence, f-divergence, Arimoto's mutual information, Sibson's information radius, and others. These new measures typically generalize the classical measures and in some applications provide finer results. In recent years they have also found new applications in guessing, hypothesis testing, error exponents, task encoding, large deviations, etc.

For this Special Issue we solicit original papers presenting new applications of known information measures and new measures with interesting applications.

Prof. Dr. Amos Lapidoth
Guest Editor

Manuscript Submission Information

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Keywords

f-divergence
Rényi Divergence
Rényi Entropy
Arimoto's Mutual Information
Information Measures
Information Radius

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Published Papers (7 papers)

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Research

17 pages, 324 KiB

Open AccessArticle

Conditional Rényi Entropy and the Relationships between Rényi Capacities

by Gautam Aishwarya and Mokshay Madiman

Entropy 2020, 22(5), 526; https://doi.org/10.3390/e22050526 - 6 May 2020

Cited by 4 | Viewed by 3011

Abstract

The analogues of Arimoto’s definition of conditional Rényi entropy and Rényi mutual information are explored for abstract alphabets. These quantities, although dependent on the reference measure, have some useful properties similar to those known in the discrete setting. In addition to laying out some such basic properties and the relations to Rényi divergences, the relationships between the families of mutual informations defined by Sibson, Augustin-Csiszár, and Lapidoth-Pfister, as well as the corresponding capacities, are explored. Full article

(This article belongs to the Special Issue Information Measures with Applications)

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59 pages, 750 KiB

Open AccessArticle

Generalizations of Fano’s Inequality for Conditional Information Measures via Majorization Theory

by Yuta Sakai

Entropy 2020, 22(3), 288; https://doi.org/10.3390/e22030288 - 1 Mar 2020

Cited by 4 | Viewed by 3963

Abstract

Fano’s inequality is one of the most elementary, ubiquitous, and important tools in information theory. Using majorization theory, Fano’s inequality is generalized to a broad class of information measures, which contains those of Shannon and Rényi. When specialized to these measures, it recovers and generalizes the classical inequalities. Key to the derivation is the construction of an appropriate conditional distribution inducing a desired marginal distribution on a countably infinite alphabet. The construction is based on the infinite-dimensional version of Birkhoff’s theorem proven by Révész [Acta Math. Hungar. 1962, 3, 188–198], and the constraint of maintaining a desired marginal distribution is similar to coupling in probability theory. Using our Fano-type inequalities for Shannon’s and Rényi’s information measures, we also investigate the asymptotic behavior of the sequence of Shannon’s and Rényi’s equivocations when the error probabilities vanish. This asymptotic behavior provides a novel characterization of the asymptotic equipartition property (AEP) via Fano’s inequality. Full article

(This article belongs to the Special Issue Information Measures with Applications)

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23 pages, 518 KiB

Open AccessArticle

Rényi and Tsallis Entropies of the Aharonov–Bohm Ring in Uniform Magnetic Fields

by Oleg Olendski

Entropy 2019, 21(11), 1060; https://doi.org/10.3390/e21111060 - 29 Oct 2019

Cited by 20 | Viewed by 3687

Abstract

One-parameter functionals of the Rényi

R_{ρ, γ} (α)

and Tsallis

T_{ρ, γ} (α)

types are calculated both in the position (subscript

ρ

) and momentum (

γ

) spaces for the azimuthally symmetric 2D nanoring [...] Read more.

One-parameter functionals of the Rényi

R_{ρ, γ} (α)

and Tsallis

T_{ρ, γ} (α)

types are calculated both in the position (subscript

ρ

) and momentum (

γ

) spaces for the azimuthally symmetric 2D nanoring that is placed into the combination of the transverse uniform magnetic field

B

and the Aharonov–Bohm (AB) flux

ϕ_{A B}

and whose potential profile is modeled by the superposition of the quadratic and inverse quadratic dependencies on the radius r. Position (momentum) Rényi entropy depends on the field B as a negative (positive) logarithm of

ω_{e f f} \equiv {(ω_{0}^{2} + ω_{c}^{2} / 4)}^{1 / 2}

, where

ω_{0}

determines the quadratic steepness of the confining potential and

ω_{c}

is a cyclotron frequency. This makes the sum

{R_{ρ}}_{n m} (α) + {R_{γ}}_{n m} (\frac{α}{2 α - 1})

a field-independent quantity that increases with the principal n and azimuthal m quantum numbers and satisfies the corresponding uncertainty relation. In the limit

α \to 1

, both entropies in either space tend to their Shannon counterparts along, however, different paths. Analytic expression for the lower boundary of the semi-infinite range of the dimensionless coefficient

α

where the momentum entropies exist reveals that it depends on the ring geometry, AB intensity, and quantum number m. It is proved that there is the only orbital for which both Rényi and Tsallis uncertainty relations turn into the identity at

α = 1 / 2

, which is not necessarily the lowest-energy level. At any coefficient

α

, the dependence of the position of the Rényi entropy on the AB flux mimics the energy variation with

ϕ_{A B}

, which, under appropriate scaling, can be used for the unique determination of the associated persistent current. Similarities and differences between the two entropies and their uncertainty relations are discussed as well. Full article

(This article belongs to the Special Issue Information Measures with Applications)

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80 pages, 1639 KiB

Open AccessFeature PaperArticle

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

by Igal Sason

Entropy 2019, 21(10), 1022; https://doi.org/10.3390/e21101022 - 21 Oct 2019

Cited by 18 | Viewed by 4688

Abstract

This paper is focused on the derivation 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 the major part of this manuscript. Full article

(This article belongs to the Special Issue Information Measures with Applications)

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14 pages, 324 KiB

Open AccessArticle

Entropic Matroids and Their Representation

by Emmanuel Abbe and Sophie Spirkl

Entropy 2019, 21(10), 948; https://doi.org/10.3390/e21100948 - 27 Sep 2019

Cited by 1 | Viewed by 3070

Abstract

This paper investigates entropic matroids, that is, matroids whose rank function is given as the Shannon entropy of random variables. In particular, we consider p-entropic matroids, for which the random variables each have support of cardinality p. We draw connections [...] Read more.

p = 2, 3

but not when

p = 9

. Our results leave open the possibility for p-entropic matroids to be linear whenever p is prime, with particular cases proved here. Applications of entropic matroids to coding theory and cryptography are also discussed. Full article

(This article belongs to the Special Issue Information Measures with Applications)

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40 pages, 955 KiB

Open AccessEditor’s ChoiceArticle

Two Measures of Dependence

by Amos Lapidoth and Christoph Pfister

Entropy 2019, 21(8), 778; https://doi.org/10.3390/e21080778 - 8 Aug 2019

Cited by 13 | Viewed by 3677

Abstract

Two families of dependence measures between random variables are introduced. They are based on the Rényi divergence of order

α

and the relative

α

-entropy, respectively, and both dependence measures reduce to Shannon’s mutual information when their order

α

is one. The first [...] Read more.

Two families of dependence measures between random variables are introduced. They are based on the Rényi divergence of order

α

and the relative

α

-entropy, respectively, and both dependence measures reduce to Shannon’s mutual information when their order

α

is one. The first measure shares many properties with the mutual information, including the data-processing inequality, and can be related to the optimal error exponents in composite hypothesis testing. The second measure does not satisfy the data-processing inequality, but appears naturally in the context of distributed task encoding. Full article

(This article belongs to the Special Issue Information Measures with Applications)

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16 pages, 308 KiB

Open AccessEditor’s ChoiceArticle

Empirical Estimation of Information Measures: A Literature Guide

by Sergio Verdú

Entropy 2019, 21(8), 720; https://doi.org/10.3390/e21080720 - 24 Jul 2019

Cited by 44 | Viewed by 10396

Abstract

We give a brief survey of the literature on the empirical estimation of entropy, differential entropy, relative entropy, mutual information and related information measures. While those quantities are of central importance in information theory, universal algorithms for their estimation are increasingly important in [...] Read more.

(This article belongs to the Special Issue Information Measures with Applications)

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