Sobolev–Kantorovich inequalities under CD(0,∞) condition

2021 ◽  
pp. 2150027
Author(s):  
Vladimir I. Bogachev ◽  
Alexander V. Shaposhnikov ◽  
Feng-Yu Wang
Keyword(s):  
Riemannian Manifold ◽  
Probability Density ◽  
Sobolev Norm ◽  
Interpolation Inequalities ◽  
Reference Measure ◽  
Kantorovich Distance ◽  
Text Condition

We refine and generalize several interpolation inequalities bounding the [Formula: see text] norm of a probability density with respect to the reference measure [Formula: see text] by its Sobolev norm and the Kantorovich distance to [Formula: see text] on a smooth weighted Riemannian manifold satisfying [Formula: see text] condition.

scholarly journals Sobolev-Kantorovich Inequalities

2015 ◽  
Vol 3 (1) ◽  
Author(s):  
Michel Ledoux
Keyword(s):  
Pattern Formation ◽  
Probability Density ◽  
Riemannian Manifolds ◽  
Wasserstein Distance ◽  
Sobolev Norm ◽  
Heat Flows ◽  
Interpolation Inequalities ◽  
Harnack Inequalities ◽  
Negatively Curved ◽  
Reference Measure

Abstract In a recent work, E. Cinti and F. Otto established some new interpolation inequalities in the study of pattern formation, bounding the Lr(μ)-norm of a probability density with respect to the reference measure μ by its Sobolev norm and the Kantorovich-Wasserstein distance to μ. This article emphasizes this family of interpolation inequalities, called Sobolev-Kantorovich inequalities, which may be established in the rather large setting of non-negatively curved (weighted) Riemannian manifolds by means of heat flows and Harnack inequalities.

Modified algorithm for fast bandwidth selection for kernel estimates of multidimensional probability densities

2020 ◽  
pp. 9-13
Author(s):  
A. V. Lapko ◽  
V. A. Lapko
Keyword(s):  
Probability Density ◽  
Optimal Parameter ◽  
Random Variables ◽  
Kernel Functions ◽  
Independent Random Variables ◽  
Functional Dependencies ◽  
Approximation Properties ◽  
Probability Density Estimation ◽  
Multidimensional Probability ◽  
Selection Of

An original technique has been justified for the fast bandwidths selection of kernel functions in a nonparametric estimate of the multidimensional probability density of the Rosenblatt–Parzen type. The proposed method makes it possible to significantly increase the computational efficiency of the optimization procedure for kernel probability density estimates in the conditions of large-volume statistical data in comparison with traditional approaches. The basis of the proposed approach is the analysis of the optimal parameter formula for the bandwidths of a multidimensional kernel probability density estimate. Dependencies between the nonlinear functional on the probability density and its derivatives up to the second order inclusive of the antikurtosis coefficients of random variables are found. The bandwidths for each random variable are represented as the product of an undefined parameter and their mean square deviation. The influence of the error in restoring the established functional dependencies on the approximation properties of the kernel probability density estimation is determined. The obtained results are implemented as a method of synthesis and analysis of a fast bandwidths selection of the kernel estimation of the two-dimensional probability density of independent random variables. This method uses data on the quantitative characteristics of a family of lognormal distribution laws.

A Delta-Sigma Modulator Using a Non-uniform Quantizer Adjusted for the Probability Density of Input Signals

2012 ◽  
Vol E95.B (7) ◽  
pp. 2257-2265
Author(s):  
Toru KITAYABU ◽  
Mao HAGIWARA ◽  
Hiroyasu ISHIKAWA ◽  
Hiroshi SHIRAI
Keyword(s):  
Probability Density ◽  
Delta Sigma Modulator ◽  
Delta Sigma ◽  
Input Signals
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