Entropy Power Inequality for the Rényi Entropy

Bobkov, Sergey G.; Chistyakov, G. P.

doi:10.1109/tit.2014.2383379

Cited by 74 publications

(90 citation statements)

References 14 publications

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“…Madiman [11] used basic information theoretic relations to prove the submodularity of the entropy of independent sums and found accordingly upper bounds on the discrete and differential entropy of sums. Though, in its general form, the problem of upper bounding the differential entropy of independent sums is not always possible (proposition 4, [2]), several results are known in particular settings. Cover et al [12] solved the problem of maximizing the differential entropy of the sum of dependent RVs having the same marginal log-concave densities.…”

Section: -mentioning

confidence: 99%

“…The EPI states that given two real independent RVs X, Z such that h(X), h(Z) and h(X + Z) exist, then (Corollary 3, [2])…”

mentioning

confidence: 99%

“…While Shannon proposed Equation (2) and proved it locally around the normal distribution, Stam [3] was the first to prove this result in general followed by Blachman [4] in what is considered to be a simplified proof. The proof was done via the usage of two information identities:…”

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confidence: 99%

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A New Tight Upper Bound on the Entropy of Sums

Fahs

Abou-Faycal

2015

Entropy

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Abstract:We consider the independent sum of a given random variable with a Gaussian variable and an infinitely divisible one. We find a novel tight upper bound on the entropy of the sum which still holds when the variable possibly has an infinite second moment. The proven bound has several implications on both information theoretic problems and infinitely divisible noise channels' transmission rates.

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Section: -mentioning

confidence: 99%

“…The EPI states that given two real independent RVs X, Z such that h(X), h(Z) and h(X + Z) exist, then (Corollary 3, [2])…”

mentioning

confidence: 99%

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A New Tight Upper Bound on the Entropy of Sums

Fahs

Abou-Faycal

2015

Entropy

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“…Remark 4.6. Recent work of Bobkov and Chistyakov [4] and of Wang and Madiman [22] has studied formulations of Shannon's Entropy Power Inequality (EPI) for Rényi entropy of the convolution of probability densities f i in R d . To be specific, [4], Theorem 1, shows that for q > 1, the Rényi entropy power N R,q satisfies…”

Section: For Q > 1 If the Rényi Entropy Is Concave Then So Is The Tmentioning

confidence: 99%

A proof of the Shepp–Olkin entropy concavity conjecture

Hillion¹,

Johnson²

2017

Bernoulli

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This is the final published version of the article (version of record). It first appeared online via Project Euclid at http://projecteuclid.org/euclid.bj/1495505104. Please refer to any applicable terms of use of the publisher. University of Bristol -Explore Bristol Research General rightsThis document is made available in accordance with publisher policies. Please cite only the published version using the reference above. Full terms of use are available: http://www.bristol.ac.uk/pure/about/ebr-terms We prove the Shepp-Olkin conjecture, which states that the entropy of the sum of independent Bernoulli random variables is concave in the parameters of the individual random variables. Our proof refines an argument previously presented by the same authors, which resolved the conjecture in the monotonic case (where all the parameters are simultaneously increasing). In fact, we show that the monotonic case is the worst case, using a careful analysis of concavity properties of the derivatives of the probability mass function. We propose a generalization of Shepp and Olkin's original conjecture, to consider Rényi and Tsallis entropies.

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“…Using this fact, the authors of [28] applied the entropy power inequality for the Shannon entropy. For the Rényi α-entropy, a version of entropy power inequalities was given in [57] but only for α ≥ 1. The problem of extending the entropy power inequality to orders 0 < α < 1 remains open.…”

Section: Entanglement Criteria For a Multipartite Quantum Systemmentioning

confidence: 99%

Rényi formulation of entanglement criteria for continuous variables

Rastegin

2017

Phys. Rev. A

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Entanglement criteria for an n-partite quantum system with continuous variables are formulated in terms of Rényi entropies. Rényi entropies are widely used as a good information measure due to many nice properties. Derived entanglement criteria are based on several mathematical results such as the Hausdorff-Young inequality, Young's inequality for convolution and its converse. From the historical viewpoint, the formulations of these results with sharp constants were obtained comparatively recently. Using the position and momentum observables of subsystems, one can build two total-system measurements with the following property. For product states, the final density in each global measurement appears as a convolution of n local densities. Hence, restrictions in terms of two Rényi entropies with constrained entropic indices are formulated for n-separable states of an n-partite quantum system with continuous variables. Experimental results are typically sampled into bins between prescribed discrete points. For these aims, we give appropriate reformulations of the derived entanglement criteria.

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Entropy Power Inequality for the Rényi Entropy

Cited by 74 publications

References 14 publications

A New Tight Upper Bound on the Entropy of Sums

A New Tight Upper Bound on the Entropy of Sums

A proof of the Shepp–Olkin entropy concavity conjecture

Rényi formulation of entanglement criteria for continuous variables

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