Some Inequalities Satisfied by the Quantities of Information of Fisher and Shannon

By A. J. Stam. 1959

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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.”
Maximum Likelihood Training of Score-Based Diffusion Models (2021, background): “Our result in Theorem 2 can be viewed as a generalization of De Bruijin’s identity ([49], Eq.”
PacGAN: The Power of Two Samples in Generative Adversarial Networks (2017, background): “, [26], [27], [28], [29], [30], [31], [32], [33], [34]) where operational interpretations of mutual information and corresponding data processing inequalities have given rise to simple proofs of strong technical results.”
Kantorovich duality for general transport costs and applications (2014, background): “1) is the Logarithmic Sobolev inequality introduced by Gross [33] (see also Stam [58] and Federbush [22]).”
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].”
Information Inequalities for Joint Distributions, With Interpretations and Applications (2008, background): “Introduced by [45] and [48] in seminal contributions, they have proved to be extremely useful and surprisingly deep—with connections to functional analysis, central limit theorems, and to the determination of capacity and rate regions for problems in information theory.”
Functional affine-isoperimetry and an inverse logarithmic Sobolev inequality (2011, background): “Logarithmic Sobolev inequalities provide upper bounds for the entropy.”
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”
On the rate of convergence in the entropic central limit theorem (2004, background): “…this identity reduces to∫ w [ Tr(Dp)2 + 〈Hess(− log w)p, p〉 ] = J + W (∫ f(r′)2 + ∫ fφ′′r2 ) + JV ∫ fr2 + J(W − V ) (∫ fr )2 +2U (∫ fφ′′r − J ∫ fr ) − 2W (∫ fr )(∫ fφ′′r ) + M (∫ fr′ )2 , where: J = I(X), U = 1− n∑ i=1 a4i , V = n∑ i=1 a4i − n∑ i=1 a6i , W = U − V and M = ( n∑ i=1 a3i )2 − n∑ i=1…”
Functional Versions of Lp-Affine Surface Area and Entropy Inequalities (2014, unknown): “The log-Sobolev inequality, due to Gross [23] (see also [17, 49]), asserts that for every probability measure µ on R n that is absolutely continuous with respect to Lebesgue measure, where H and I denote the relative entropy and Fisher information, respectively, and | · | is the Euclidean norm.”
On the trend to global equilibrium in spatially inhomogeneous entropy‐dissipating systems: The linear Fokker‐Planck equation (2001, background): “Starting at least from the work of Bakry and Emery fifteen years ago [4], it was understood that explicit estimates of relaxation to equilibrium could be obtained by the direct study of the dissipation of entropy, with the help of so-called logarithmic Sobolev inequalities [32, 19, 4, 20, 33].”
Contractive Probability Metrics and Asymptotic Behavior of Dissipative Kinetic Equations (2007, background): “This is a consequence of Shannon’s entropy power inequality [99, 16].”
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.”
An asymmetric affine Pólya–Szegö principle (2012, background): “However, it is well known that inequality (44) is equivalent to the logarithmic Sobolev inequality with respect to Gauss measure due to Stam [64] and Gross [34].”
Gradient of mutual information in linear vector Gaussian channels (2006, background): “One common example is the design of a transmit precoder for a given communication system; in particular, the precoder can be flexible and adapt to the channel realization to increase the system performance [1].”
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 ].”
Entropy jumps for isotropic log-concave random vectors and spectral gap (2012, background): “The celebrated Shannon–Stam inequality (see [22, 23]) says that if X and Y are independent identically distributed (i.”