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.

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.

The free logarithmic Sobolev and the free transportation cost inequalities by time integrations

2014 ◽  
Vol 17 (03) ◽  
pp. 1450022 ◽  
Author(s):  
Aya Nemoto ◽  
Hiroaki Yoshida
Keyword(s):  
Time Derivative ◽  
Planck Equation ◽  
Transportation Cost ◽  
Wasserstein Distance ◽  
Sobolev Inequalities ◽  
Differential Inequalities ◽  
Mass Transportation ◽  
Logarithmic Sobolev Inequalities ◽  
One Dimensional ◽  
Free Entropy

In this paper, we shall revisit the free analogues of the logarithmic Sobolev and the transportation cost inequalities for one-dimensional case by time integrations. We consider time evolutions by the free Fokker–Planck equation and calculate the time derivative of the 2-Wasserstein distance with the optimal mass transportation, from which some differential inequalities can be derived. The convergence to the equilibrium in the relative free entropy is discussed, and the free transportation cost and the free logarithmic Sobolev inequalities can be obtained by time integrations.

scholarly journals On the Geodesic Distance in Shapes K-means Clustering

Entropy ◽  
2018 ◽  
Vol 20 (9) ◽  
pp. 647 ◽  
Author(s):  
Stefano Gattone ◽  
Angela De Sanctis ◽  
Stéphane Puechmorel ◽  
Florence Nicol
Keyword(s):  
Complex Shape ◽  
Geodesic Distance ◽  
Transportation Cost ◽  
Information Geometry ◽  
Wasserstein Distance ◽  
Discriminative Power ◽  
Statistical Manifold ◽  
Probability Densities ◽  
Different Shapes ◽  
Rotationally Invariant

In this paper, the problem of clustering rotationally invariant shapes is studied and a solution using Information Geometry tools is provided. Landmarks of a complex shape are defined as probability densities in a statistical manifold. Then, in the setting of shapes clustering through a K-means algorithm, the discriminative power of two different shapes distances are evaluated. The first, derived from Fisher–Rao metric, is related with the minimization of information in the Fisher sense and the other is derived from the Wasserstein distance which measures the minimal transportation cost. A modification of the K-means algorithm is also proposed which allows the variances to vary not only among the landmarks but also among the clusters.

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