A Free Logarithmic Sobolev Inequality on the Circle

2006 ◽  
Vol 49 (3) ◽  
pp. 389-406 ◽  
Author(s):  
Fumio Hiai ◽  
Dénes Petz ◽  
Yoshimichi Ueda
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Large Deviation ◽  
Logarithmic Sobolev Inequality ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Unitary Matrices ◽  
Free Entropy ◽  
Large Deviation Result ◽  
Free Fisher Information

AbstractFree analogues of the logarithmic Sobolev inequality compare the relative free Fisher information with the relative free entropy. In the present paper such an inequality is obtained for measures on the circle. The method is based on a random matrix approximation procedure, and a large deviation result concerning the eigenvalue distribution of special unitary matrices is applied and discussed.

REMARKS ON THE FREE RELATIVE ENTROPY AND THE FREE FISHER INFORMATION DISTANCE

2010 ◽  
Vol 13 (02) ◽  
pp. 243-255
Author(s):  
HIROAKI YOSHIDA
Keyword(s):  
Relative Entropy ◽  
Fisher Information ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Semicircle Law ◽  
Information Distance ◽  
Poisson Law ◽  
Free Relative ◽  
Free Fisher Information

In this paper, we shall introduce the free Fisher information distance which is inspired by the estimation-theoretic representation of the free relative entropy investigated by Verdú. We shall see the free analogue of the logarithmic Sobolev inequality with respect to a centered semicircle law and also the semicircular approximation of the free Poisson law.

scholarly journals Skewed Jensen—Fisher Divergence and Its Bounds

Foundations ◽  
2021 ◽  
Vol 1 (2) ◽  
pp. 256-264
Author(s):  
Takuya Yamano
Keyword(s):  
Probability Density ◽  
Fisher Information ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Density Functions ◽  
Density Functions ◽  
De Bruijn ◽  
Measure Of Similarity ◽  
Jensen Shannon Divergence ◽  
Variational Distance

A non-uniform (skewed) mixture of probability density functions occurs in various disciplines. One needs a measure of similarity to the respective constituents and its bounds. We introduce a skewed Jensen–Fisher divergence based on relative Fisher information, and provide some bounds in terms of the skewed Jensen–Shannon divergence and of the variational distance. The defined measure coincides with the definition from the skewed Jensen–Shannon divergence via the de Bruijn identity. Our results follow from applying the logarithmic Sobolev inequality and Poincaré inequality.

scholarly journals Fisher Information and Logarithmic Sobolev Inequality for Matrix-Valued Functions

2020 ◽  
Vol 21 (11) ◽  
pp. 3409-3478
Author(s):  
Li Gao ◽  
Marius Junge ◽  
Nicholas LaRacuente
Keyword(s):  
Fisher Information ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality

REMARKS ON A SEMICIRCULAR PERTURBATION OF THE FREE FISHER INFORMATION

2008 ◽  
Vol 11 (01) ◽  
pp. 97-108 ◽  
Author(s):  
HIROAKI YOSHIDA
Keyword(s):  
Fisher Information ◽  
Conditional Variance ◽  
Information Inequality ◽  
Free Entropy ◽  
Variance Formula ◽  
The Mean ◽  
Power Inequality ◽  
Free Fisher Information ◽  
Special Case

In this paper, we shall give the representation of a semicircular perturbation of the free Fisher information [Formula: see text] by the mean of the conditional variance [Formula: see text], where S is a standard semicircular element freely independent of X, and ε > 0. Using this representation, we will give alternative proofs of the free Fisher information inequality, in which the free analogue of Stam's inequality can be obtained as a special case, and of the free entropy power inequality in an infinitesimal approach to the free entropy.

scholarly journals FREE TRANSPORTATION COST INEQUALITIES FOR NONCOMMUTATIVE MULTI-VARIABLES

2006 ◽  
Vol 09 (03) ◽  
pp. 391-412 ◽  
Author(s):  
FUMIO HIAI ◽  
YOSHIMICHI UEDA
Keyword(s):  
Transportation Cost ◽  
Wasserstein Distance ◽  
Approximation Procedure ◽  
Matrix Approximation ◽  
Convexity Condition ◽  
Free Entropy ◽  
Additive Type ◽  
Free Case ◽  
Probability Spaces ◽  
Transportation Cost Inequality

The free analogue of the transportation cost inequality so far obtained for measures is extended to the noncommutative setting. Our free transportation cost inequality is for tracial distributions of noncommutative self-adjoint (also unitary) multi-variables in the framework of tracial C*-probability spaces, and it tells that the Wasserstein distance is dominated by the square root of the relative free entropy with respect to a potential of additive type (corresponding to the free case) with some convexity condition. The proof is based on random matrix approximation procedure.

scholarly journals Functional Inequalities for Two-Level Concentration

2021 ◽  
Author(s):  
Franck Barthe ◽  
Michał Strzelecki
Keyword(s):  
Sobolev Inequality ◽  
Poincaré Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Probability Measures ◽  
Exponential Rate ◽  
Functional Inequalities ◽  
Concentration Inequality ◽  
Poincare Inequality ◽  
Free Concentration ◽  
Concentration Property

AbstractProbability measures satisfying a Poincaré inequality are known to enjoy a dimension-free concentration inequality with exponential rate. A celebrated result of Bobkov and Ledoux shows that a Poincaré inequality automatically implies a modified logarithmic Sobolev inequality. As a consequence the Poincaré inequality ensures a stronger dimension-free concentration property, known as two-level concentration. We show that a similar phenomenon occurs for the Latała–Oleszkiewicz inequalities, which were devised to uncover dimension-free concentration with rate between exponential and Gaussian. Motivated by the search for counterexamples to related questions, we also develop analytic techniques to study functional inequalities for probability measures on the line with wild potentials.

Equivalence between logarithmic Sobolev inequality and hypercontractivity in a probability gage space

2020 ◽  
pp. 1-13
Author(s):  
Zhang Lunchuan
Keyword(s):  
Dirichlet Form ◽  
Sobolev Inequality ◽  
Logarithmic Sobolev Inequality ◽  
Markov Semigroup ◽  
Quantum Markov Semigroup

Abstract In this paper, we prove the equivalence between logarithmic Sobolev inequality and hypercontractivity of a class of quantum Markov semigroup and its associated Dirichlet form based on a probability gage space.

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