A concavity property for the reciprocal of Fisher information and its consequences on Costa’s EPI

Toscani, Giuseppe

doi:10.1016/j.physa.2015.03.018

Cited by 16 publications

(15 citation statements)

References 22 publications

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“…Notice that the third derivative of the entropy power N(X + √ tZ) was shown to be nonnegative under the log-concavity condition [5], and we recover this in Corollary 3. We also considered the fourth derivative, but failed to obtain the sign because we were unable to apply the Cauchy-Schwartz inequality as we did for the third derivative.…”

Section: On the Derivativessupporting

confidence: 63%

“…As a corollary, we recover Toscani's result [5] on the third derivative of the entropy power, using the Cauchy-Schwartz inequality, which is much simpler. In Section 3, we introduce the linear matrix inequality approach, and transform the above two conjectures to the feasibility check of semidefinite programming problems.…”

Section: Introductionsupporting

confidence: 51%

“…If the probability density function of X + √ tZ is log-concave, then the fifth derivative of the differential entropy is strictly positive: h (5) …”

Section: Corollarymentioning

confidence: 99%

“…Instead, if we consider the log-concavity constraint T 2 ≤ 0 and check the optimality of Gaussian inputs, then we have a new matrixF 0 (similar to Section 3.5) and several matrices G j as the following: 1), (1, 3), (1,5), (2,3), (2,5), (3,3), (3,4), (3,5), (3,6), (5,5), (5,6) …”

Section: Fifth Derivativementioning

confidence: 99%

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Gaussian Optimality for Derivatives of Differential Entropy Using Linear Matrix Inequalities

Zhang

Anantharam

Geng

2018

Entropy

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Let Z be a standard Gaussian random variable, X be independent of Z, and t be a strictly positive scalar. For the derivatives in t of the differential entropy of X + √ tZ, McKean noticed that Gaussian X achieves the extreme for the first and second derivatives, among distributions with a fixed variance, and he conjectured that this holds for general orders of derivatives. This conjecture implies that the signs of the derivatives alternate. Recently, Cheng and Geng proved that this alternation holds for the first four orders. In this work, we employ the technique of linear matrix inequalities to show that: firstly, Cheng and Geng's method may not generalize to higher orders; secondly, when the probability density function of X + √ tZ is log-concave, McKean's conjecture holds for orders up to at least five. As a corollary, we also recover Toscani's result on the sign of the third derivative of the entropy power of X + √ tZ, using a much simpler argument.

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Section: On the Derivativessupporting

confidence: 63%

Section: Introductionsupporting

confidence: 51%

“…If the probability density function of X + √ tZ is log-concave, then the fifth derivative of the differential entropy is strictly positive: h (5) …”

Section: Corollarymentioning

confidence: 99%

Section: Fifth Derivativementioning

confidence: 99%

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Gaussian Optimality for Derivatives of Differential Entropy Using Linear Matrix Inequalities

Zhang

Anantharam

Geng

2018

Entropy

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show abstract

“…Log-concave random vectors and functions are important classes in many disciplines. In the context of information theory, several nice properties involving entropy of log-concave random vectors were recently established (see, e.g., [3,5,18,33,40,41]). Significant examples are Gaussian and exponential distributions as well as any uniform distribution on a convex set.…”

Section: Preliminariesmentioning

confidence: 99%