Related papers: Forward and Reverse Entropy Power Inequalities in …
We survey some interplays between spectral estimates of H\"ormander-type, degenerate Monge-Amp\`ere equations and geometric inequalities related to log-concavity such as Brunn-Minkowski, Santal\'o or Busemann inequalities.
In this paper, I expand Shannon's definition of entropy into a new form of entropy that allows integration of information from different random events. Shannon's notion of entropy is a special case of my more general definition of entropy.…
This paper deals with the famous isoperimetric inequality. In a first part, we give some new functional form of the isoperimetric inequality, and in a second part, we give a quantitative form with a remainder term involving Wasserstein…
The weak law of large numbers implies that, under mild assumptions on the source, the Renyi entropy per produced symbol converges (in probability) towards the Shannon entropy rate. This paper quantifies the speed of this convergence for…
In this note we show how to construct a number of nonconvex quadratic inequalities for a variety of physics equations appearing in physical design problems. These nonconvex quadratic inequalities can then be used to construct bounds on…
Mixture distributions are extensively used as a modeling tool in diverse areas from machine learning to communications engineering to physics, and obtaining bounds on the entropy of probability distributions is of fundamental importance in…
Quantum coherence and entanglement are two key features in quantum mechanics and play important roles in quantum information processing and quantum computation. We provide a general triangle-like inequality satisfied by the $l_1$-norm…
Using convex Grothendieck fibrations, we characterize the von Neumann entropy as a functor from finite-dimensional non-commutative probability spaces and state-preserving *-homomorphisms to real numbers. Our axioms reproduce those of Baez,…
Functional inequalities such as the Poincar\'e and log-Sobolev inequalities quantify convergence to equilibrium in continuous-time Markov chains by linking generator properties to variance and entropy decay. However, many applications,…
Many natural and social science systems are described using probability distributions over elements that are related to each other: for instance, occupations with shared skills or species with similar traits. Standard information theory…
Entropy and relative or cross entropy measures are two very fundamental concepts in information theory and are also widely used for statistical inference across disciplines. The related optimization problems, in particular the maximization…
A number of inequalities for the weighted entropies is proposed, mirroring properties of a standard (Shannon) entropy and related quantities.
On a Riemannian manifold, lower Ricci curvature bounds are known to be characterized by geodesic convexity properties of various entropies with respect to the Kantorovich-Rubinstein-Wasserstein square distance from optimal transportation.…
We show that Shannon's entropy--power inequality admits a strengthened version in the case in which the densities are log-concave. In such a case, in fact, one can extend the Blachman--Stam argument to obtain a sharp inequality for the…
We show that Lorentz-Finsler geometry offers a powerful tool in obtaining inequalities. With this aim, we first point out that a series of famous inequalities such as: the (weighted) arithmetic-geometric mean inequality, Acz\'el's,…
In the estimation theory context, we generalize the notion of Shannon's entropy power to the R\'{e}nyi-entropy setting. This not only allows to find new estimation inequalities, such as the R\'{e}nyi-entropy based De Bruijn identity,…
This paper represents an extended version of an earlier note [10]. The concept of weighted entropy takes into account values of different outcomes, i.e., makes entropy context-dependent, through the weight function. We analyse analogs of…
We investigate the effect of different metrizations of probability spaces on the information geometric complexity of entropic motion on curved statistical manifolds. Specifically, we provide a comparative analysis based upon Riemannian…
The quantum version of a fundamental entropic data-processing inequality is presented. It establishes a lower bound for the entropy that can be generated in the output channels of a scattering process, which involves a collection of…
A number of recent works have sought to generalize the Kolmogorov-Sinai entropy of probability-preserving transformations to the setting of Markov operators acting on the integrable functions on a probability space $(X,\mu)$. These have…