The convolution inequality for entropy powers

By N. Blachman. 1965

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Citations in the corpus

The Composition Theorem for Differential Privacy (2013, neutral): “Comparing to (4), the privacy guarantee is signiﬁcantly improved from ˜ ε ˜ δ = O kε 2 + kε 2 log(1 / ˜ δ ) to ˜ ε ˜ δ = O kε 2 + min kε 2 log(1 / ˜ δ ) , ε log( ε/ ˜ δ ) (cid:9)(cid:17) , especially when composing a large number k of interactive queries.”
Entropy jumps for isotropic log-concave random vectors and spectral gap (2012, background): “mal) Gaussian vector with density g, then the Pinsker–Csiszár–Kullback inequality (see [20, 11] or [5]) implies that 1 2 (”
Information theoretic inequalities (1991, methodology): “This equivalent statement of the Fisher information inequality is proved for example in [6] (for n = 11 and [20] (for y1# 1).”
Chapter 2 – A Review of Mathematical Topics in Collisional Kinetic Theory (2002, unknown): “The famous BlachmanStam and Shannon-Stam inequalities(11) [411, 75, 125] admit as particular cases I(f ? f) ≤ I(f), H(f ? f) ≤ H(f).”
Fisher information inequalities and the central limit theorem (2001, background): “Equation ( 3 ) to (1 − β) times Equation (4), we deduce the result.”
The Capacity Region of the Gaussian Multiple-Input Multiple-Output Broadcast Channel (2006, methodology): “Instead, the entropy power inequality (EPI) [1], [3] is used.”
Gradient of mutual information in linear vector Gaussian channels (2006, background): “Matrix Generalization of De Bruijn’s Identity For a multivariate density function , De Bruijn’s identity [2], [17]–[19] relates the derivative of the differential entropy to the Fisher information matrix defined as”
On the rate of convergence in the entropic central limit theorem (2004, background): “Note that the best choice of p would give equality in (4), whereas the simplest choice (p(x) = a for all x) recovers I ( ∑n i=1 aiXi) ≤ I(X), a consequence of the Blachman-Stam inequality [6, 15].”
An Extremal Inequality Motivated by Multiterminal Information Theoretic Problems (2006, methodology): “We oﬀered two proofs to the new extremal inequality: one by appropriately using the classical EPI, and the other by the perturbation approach of Stam [2] and Blachman [3].”
Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations (2007, background): “This is a consequence of Shannon’s entropy power inequality [99, 16].”
Capacity bounds for Cooperative diversity (2006, methodology): “We can now use the (conditional) entropy power inequality [51], [37, Theorem 16.”
Log-concavity and the maximum entropy property of the Poisson distribution (2006, methodology): “As in Stam [20] and Blachman [3], we will differentiate along this family of maps, and see that the resulting probabilities satisfy a partial differential-difference equation.”
Some results concerning maximum Renyi entropy distributions (2005, unknown): “More rigorous proofs appeared in Blachman [3] and later in Dembo, Cover and Thomas [8].”
ON CONVEX SOBOLEV INEQUALITIES AND THE RATE OF CONVERGENCE TO EQUILIBRIUM FOR FOKKER-PLANCK TYPE EQUATIONS (2001, methodology): “In the same paper, the connection between Gross’ logarithmic Sobolev inequality and the linear Fokker-Planck equation (with quadratic confinement potential and the identity as diffusion matrix) , through the Fisher measure of information and the Blachman-Stam inequality [ Bla65 ], [Sta59] has been f”
The Wasserstein Gradient Flow of the Fisher Information and the Quantum Drift-diffusion Equation (2009, methodology): “In [8] Blachman presented a simple proof of Shannon’s convolution inequality previously obtained by Stam [44], showing convolution inequalities for Fisher information, and subsequently for the logarithmic entropy using their link in terms of the heat kernel.”
Entropy jumps in the presence of a spectral gap (2003, methodology): “For random variables with a given variance, the Gaussian has the smallest Fisher information, and according to the Blachman-Stam inequality (see [ 3 ], [21]), the Fisher information of a general density decreases with repeated convolution.”
Cercignani's Conjecture is Sometimes True and Always Almost True (2003, methodology): “This is the point where we shall make use of the Blachman-Stam inequality [ 7 , 11, 36].”
An Extremal Inequality Motivated by Multiterminal Information-Theoretic Problems (2007, methodology): “We offered two proofs to the new extremal inequality: one by appropriately using the classical EPI, and the other by the perturbation approach of Stam [2] and Blachman [3].”