8,552 research outputs found
Yet Another Proof of the Entropy Power Inequality
Yet another simple proof of the entropy power inequality is given, which
avoids both the integration over a path of Gaussian perturbation and the use of
Young's inequality with sharp constant or R\'enyi entropies. The proof is based
on a simple change of variables, is formally identical in one and several
dimensions, and easily settles the equality case
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-
R\'enyi Entropy Power Inequalities via Normal Transport and Rotation
Following a recent proof of Shannon's entropy power inequality (EPI), a
comprehensive framework for deriving various EPIs for the R\'enyi entropy is
presented that uses transport arguments from normal densities and a change of
variable by rotation. Simple arguments are given to recover the previously
known R\'enyi EPIs and derive new ones, by unifying a multiplicative form with
constant c and a modification with exponent {\alpha} of previous works. In
particular, for log-concave densities, we obtain a simple transportation proof
of a sharp varentropy bound.Comment: 17 page. Entropy Journal, to appea
The information-theoretic meaning of Gagliardo--Nirenberg type inequalities
Gagliardo--Nirenberg inequalities are interpolation inequalities which were
proved independently by Gagliardo and Nirenberg in the late fifties. In recent
years, their connections with theoretic aspects of information theory and
nonlinear diffusion equations allowed to obtain some of them in optimal form,
by recovering both the sharp constants and the explicit form of the optimizers.
In this note, at the light of these recent researches, we review the main
connections between Shannon-type entropies, diffusion equations and a class of
these inequalities
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