The convolution inequality for entropy powers

Blachman, Nelson M.

doi:10.1109/tit.1965.1053768

Cited by 296 publications

(358 citation statements)

References 2 publications

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“…We first provide a direct proof of this relation, and then use it to unify and simplify existing proofs of the EPI via FI and via MMSE. In particular, two essential ingredients, namely, Fisher's information inequality [7], [12], and a related inequality for MMSE [9], [10], will be shown to be equivalent from (10).…”

Section: Proofs Of the Epi Via Fi And Mmse Revisitedmentioning

confidence: 99%

“…The FI representation (25) can be used similarly, yielding essentially Stam's proof of the EPI [6], [7]. These proofs are sketched below.…”

Section: Equivalent Integral Representations Of Differential Entropymentioning

confidence: 99%

“…The first rigorous proof of the EPI was given by Stam [6] (see also Blachman [7]). It is based on the properties of Fisher's information (FI)…”

Section: Introductionmentioning

confidence: 99%

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A Simple Proof of the Entropy-Power Inequality via Properties of Mutual Information

Rioul

2007

2007 IEEE International Symposium on Information Theory

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Abstract-While most useful information theoretic inequalities can be deduced from the basic properties of entropy or mutual information, Shannon's entropy power inequality (EPI) seems to be an exception: available information theoretic proofs of the EPI hinge on integral representations of differential entropy using either Fisher's information (FI) or minimum mean-square error (MMSE). In this paper, we first present a unified view of proofs via FI and MMSE, showing that they are essentially dual versions of the same proof, and then fill the gap by providing a new, simple proof of the EPI, which is solely based on the properties of mutual information and sidesteps both FI or MMSE representations.

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Section: Proofs Of the Epi Via Fi And Mmse Revisitedmentioning

confidence: 99%

“…The FI representation (25) can be used similarly, yielding essentially Stam's proof of the EPI [6], [7]. These proofs are sketched below.…”

Section: Equivalent Integral Representations Of Differential Entropymentioning

confidence: 99%

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A Simple Proof of the Entropy-Power Inequality via Properties of Mutual Information

Rioul

2007

2007 IEEE International Symposium on Information Theory

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“…This concavity has a classical analogue, first proved by Costa in [8]. Later, Dembo [12,11] simplified the proof, by an argument based on the so-called Blachman-Stam inequality [4]. More recently, Villani [44] gave a direct proof of the same inequality.…”

Section: Proofs Of Theorem and Theoremmentioning

confidence: 92%

Contractivity properties of a quantum diffusion semigroup

Datta

Pautrat

Rouzé

2017

Journal of Mathematical Physics

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We consider a quantum generalization of the classical heat equation, and study contractivity properties of its associated semigroup. We prove a Nash inequality and a logarithmic Sobolev inequality. The former leads to an ultracontractivity result. This in turn implies that the largest eigenvalue and the purity of a state with positive Wigner function, evolving under the action of the semigroup, decrease at least inverse polynomially in time, while its entropy increases at least logarithmically in time.

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“…This technique can be 1948). There are some known relations that connect the two information concepts [29][30][31]. Shannon's entropy can be, but is not always, the thermodynamic, Boltzmann entropy [28].…”

Section: Introductionmentioning

confidence: 99%

A Novel Algorithm for Image Thresholding Using Non-Parametric Fisher Information

Abdel-Azim¹,

Abo-Eleneen²,

Arabia³

2014

Proceedings of 1st International Electronic Conference on Entropy and Its Applications

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Abstract:The Fisher information (FI) measure is an important concept in statistical estimation theory and information theory. However, it has received relatively little consideration in image processing. In this paper, a novel algorithm is developed based on the nonparametric FI measure. The proposed algorithm determines the optimal threshold based on the FI measure by maximizing the measure of the separability of the resultant classes over all of the gray levels. The algorithm is compared with several classic thresholding methods on a variety of images, including some nondestructive testing (NDT) images and text document images. The experimental results show the effectiveness of the new method.

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The convolution inequality for entropy powers

Cited by 296 publications

References 2 publications

A Simple Proof of the Entropy-Power Inequality via Properties of Mutual Information

A Simple Proof of the Entropy-Power Inequality via Properties of Mutual Information

Contractivity properties of a quantum diffusion semigroup

A Novel Algorithm for Image Thresholding Using Non-Parametric Fisher Information

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