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Shannon Entropy: Mathematical View

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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: 31 August 2024 | Viewed by 4894

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Special Issue Editors

Prof. Dr. Josip Pečarić

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Guest Editor

Croatian Academy of Sciences and Arts, 10000 Zagreb, Croatia
Interests: mathematical inequalities; theory of convexity; information theory; operator theory; majorization theory; statistics; functional analysis

Dr. Muhammad Adil Khan

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Department of Mathematics, University of Peshawar, Peshawar 25120, Pakistan
Interests: mathematical inequalities; theory of convexity; information theory; operator theory; majorization theory; functional analysis

Dr. Đilda Pečarić

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Department of Media and Communication, University North, 48000 Koprivnica, Croatia
Interests: scholarly communication; bibliometrics; information theory; documentation; research methods

Special Issue Information

Dear Colleagues,

The field of mathematical inequalities and their applications has recorded exponential and significant growth in the last few decades, with a considerable impact in various areas of science, such as engineering, qualitative theory of differential and integral equations, economics, computer science, mathematical statistics, information theory etc. It is noteworthy that many innovative ideas about mathematical inequalities and their applications in various areas of science can be developed by convexity. The notion of classical convexity has been streamlined by mathematical inequalities.

Information theory is the science of information, which scientifically deals with the storage, quantification and communication of information. It emerged from Claude Shannon by considering stochastic processes as a source of information. Some of the foremost quantities of information theory are the entropy, relative entropy, Zipf–Mandelbrot entropy, mutual information, Csiszar divergence, etc., which are defined as functionals of probability distributions. In turn, they characterize the behavior of long sequences of random variables and allow us to estimate the probabilities of rare events. Information theory has extensive role in communication systems with deep connections to diverse topics such as artificial intelligence, statistical mechanics, biological evolution, statistical physics, etc.

The theory of convexity and several mathematical inequalities (majorization, Jensen’s and Levinson’s inequalities, etc.) are utilized to obtain some important inequalities for various entropies (e.g., Shannon and Zipf–Mandelbrot entropies) and for divergences (e.g., Csiszar and Kullback–Leibler divergences) in information theory. As some fundamental tools, these inequalities can be used to provide error bounds while estimating the aforesaid entropies and divergences.

This Special Issue aims to provide a forum for the presentation of new and improved techniques for obtaining inequalities and their applications in information theory. In particular, this Issue will consider original and high-quality papers pertaining to inequalities for Shannon entropy, Zipf–Mandelbrot entropy, Csiszar, Jeffrey’s and Kullback–Leibler divergences, and other related topics.

Prof. Dr. Josip Pečarić
Dr. Muhammad Adil Khan
Dr. Đilda Pečarić
Guest Editors

Manuscript Submission Information

Manuscripts should be submitted online at www.mdpi.com by registering and logging in to this website. Once you are registered, click here to go to the submission form. Manuscripts can be submitted until the deadline. All submissions that pass pre-check are peer-reviewed. Accepted papers will be published continuously in the journal (as soon as accepted) and will be listed together on the special issue website. Research articles, review articles as well as short communications are invited. For planned papers, a title and short abstract (about 100 words) can be sent to the Editorial Office for announcement on this website.

Submitted manuscripts should not have been published previously, nor be under consideration for publication elsewhere (except conference proceedings papers). All manuscripts are thoroughly refereed through a single-blind peer-review process. A guide for authors and other relevant information for submission of manuscripts is available on the Instructions for Authors page. Entropy is an international peer-reviewed open access monthly journal published by MDPI.

Please visit the Instructions for Authors page before submitting a manuscript. The Article Processing Charge (APC) for publication in this open access journal is 2600 CHF (Swiss Francs). Submitted papers should be well formatted and use good English. Authors may use MDPI's English editing service prior to publication or during author revisions.

Keywords

inequalities
theory of convexity
information theory
Shannon and Zipf–Mandelbrot entropies
divergences
applications

Published Papers (4 papers)

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Research

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16 pages, 1608 KiB

Open AccessFeature PaperArticle

Quantum Knowledge in Phase Space

by Davi Geiger

Entropy 2023, 25(8), 1227; https://doi.org/10.3390/e25081227 - 17 Aug 2023

Viewed by 1023

Abstract

Quantum physics through the lens of Bayesian statistics considers probability to be a degree of belief and subjective. A Bayesian derivation of the probability density function in phase space is presented. Then, a Kullback–Liebler divergence in phase space is introduced to define interference and entanglement. Comparisons between each of these two quantities and the entropy are made. A brief presentation of entanglement in phase space to the spin degree of freedom and an extension to mixed states completes the work. Full article

(This article belongs to the Special Issue Shannon Entropy: Mathematical View)

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29 pages, 443 KiB

Open AccessFeature PaperArticle

Capacity-Achieving Input Distributions of Additive Vector Gaussian Noise Channels: Even-Moment Constraints and Unbounded or Compact Support

by Jonah Eisen, Ravi R. Mazumdar and Patrick Mitran

Entropy 2023, 25(8), 1180; https://doi.org/10.3390/e25081180 - 08 Aug 2023

Cited by 1 | Viewed by 828

Abstract

We investigate the support of a capacity-achieving input to a vector-valued Gaussian noise channel. The input is subjected to a radial even-moment constraint and is either allowed to take any value in

R^{n}

or is restricted to a given compact subset of [...] Read more.

R^{n}

or is restricted to a given compact subset of

R^{n}

. It is shown that the support of the capacity-achieving distribution is composed of a countable union of submanifolds, each with a dimension of

n - 1

or less. When the input is restricted to a compact subset of

R^{n}

, this union is finite. Finally, the support of the capacity-achieving distribution is shown to have Lebesgue measure 0 and to be nowhere dense in

R^{n}

. Full article

(This article belongs to the Special Issue Shannon Entropy: Mathematical View)

27 pages, 381 KiB

Open AccessArticle

Uniform Treatment of Integral Majorization Inequalities with Applications to Hermite-Hadamard-Fejér-Type Inequalities and f-Divergences

by László Horváth

Entropy 2023, 25(6), 954; https://doi.org/10.3390/e25060954 - 19 Jun 2023

Cited by 1 | Viewed by 1390

Abstract

In this paper, we present a general framework that provides a comprehensive and uniform treatment of integral majorization inequalities for convex functions and finite signed measures. Along with new results, we present unified and simple proofs of classical statements. To apply our results, we deal with Hermite-Hadamard-Fejér-type inequalities and their refinements. We present a general method to refine both sides of Hermite-Hadamard-Fejér-type inequalities. The results of many papers on the refinement of the Hermite-Hadamard inequality, whose proofs are based on different ideas, can be treated in a uniform way by this method. Finally, we establish a necessary and sufficient condition for when a fundamental inequality of f-divergences can be refined by another f-divergence. Full article

(This article belongs to the Special Issue Shannon Entropy: Mathematical View)

Review

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29 pages, 505 KiB

Open AccessReview

An Information Theoretic Condition for Perfect Reconstruction

by Idris Delsol , Olivier Rioul , Julien Béguinot, Victor Rabiet and Antoine Souloumiac

Entropy 2024, 26(1), 86; https://doi.org/10.3390/e26010086 - 19 Jan 2024

Cited by 1 | Viewed by 734

Abstract

A new information theoretic condition is presented for reconstructing a discrete random variable X based on the knowledge of a set of discrete functions of X. The reconstruction condition is derived from Shannon’s 1953 lattice theory with two entropic metrics of Shannon and Rajski. Because such a theoretical material is relatively unknown and appears quite dispersed in different references, we first provide a synthetic description (with complete proofs) of its concepts, such as total, common, and complementary information. The definitions and properties of the two entropic metrics are also fully detailed and shown to be compatible with the lattice structure. A new geometric interpretation of such a lattice structure is then investigated, which leads to a necessary (and sometimes sufficient) condition for reconstructing the discrete random variable X given a set

{X_{1}, \dots, X_{n}}

of elements in the lattice generated by X. Intuitively, the components

X_{1}, \dots, X_{n}

of the original source of information X should not be globally “too far away” from X in the entropic distance in order that X is reconstructable. In other words, these components should not overall have too low of a dependence on X; otherwise, reconstruction is impossible. These geometric considerations constitute a starting point for a possible novel “perfect reconstruction theory”, which needs to be further investigated and improved along these lines. Finally, this condition is illustrated in five specific examples of perfect reconstruction problems: the reconstruction of a symmetric random variable from the knowledge of its sign and absolute value, the reconstruction of a word from a set of linear combinations, the reconstruction of an integer from its prime signature (fundamental theorem of arithmetic) and from its remainders modulo a set of coprime integers (Chinese remainder theorem), and the reconstruction of the sorting permutation of a list from a minimal set of pairwise comparisons. Full article

(This article belongs to the Special Issue Shannon Entropy: Mathematical View)

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