Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
TLDR
In this paper, it was shown that transport inequalities, similar to the one derived by M. Talagrand (1996, Geom. Funct. Anal. 6, 587-600) for the Gaussian measure, are implied by logarithmic Sobolev inequalities.About:
This article is published in Journal of Functional Analysis.The article was published on 2000-06-01 and is currently open access. It has received 1080 citations till now. The article focuses on the topics: Sobolev inequality & Interpolation inequality.read more
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Some inequalities on Riemannian manifolds linking Entropy,Fisher information, Stein discrepancy and Wasserstein distance
TL;DR: For a complete connected Riemannian manifold M, this article derived inequalities for probability measures on M linking relative entropy, Fisher information, Stein discrepancy and Wasserstein distance, which strengthened in particular the famous log-Sobolev and transportation-cost inequality and extended the so-called Entropy/Stein Discrepancy/Information (HSI) inequality established by Ledoux, Nourdin and Peccati (2015) for the standard Gaussian measure on Euclidean space to the setting of RiemANNian manifold.
Proceedings Article
Deep Diffusion-Invariant Wasserstein Distributional Classification
TL;DR: DeepWDC can substantially enhance the accuracy of several baseline deterministic classification methods and outperforms state-of-the-art-methods on 2D and 3D data containing various types of perturbations.
Journal ArticleDOI
Arbeitsgemeinschaft: Optimal Transport and Geometry
Felix Otto,Karl-Theodor Sturm +1 more
TL;DR: Optimal transport is a variational problem where one minimizes a transportation cost when transporting one density into another as mentioned in this paper, and the solution of this problem connects with convex analysis.
BookDOI
Diffusion, optimal transport and Ricci curvature
TL;DR: In this article, the link between optimal transport and Ricci curvature in smooth Riemannian geometry has been deeply studied, which has led to essentially equivalent definitions and to a nice geometric framework suitable for deep analytic results.
References
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Measure theory and fine properties of functions
TL;DR: In this article, the authors define and define elementary properties of BV functions, including the following: Sobolev Inequalities Compactness Capacity Quasicontinuity Precise Representations of Soboleve Functions Differentiability on Lines BV Function Differentiability and Structure Theorem Approximation and Compactness Traces Extensions Coarea Formula for BV Functions isoperimetric inequalities The Reduced Boundary The Measure Theoretic Boundary Gauss-Green Theorem Pointwise Properties this article.
Journal ArticleDOI
Polar Factorization and Monotone Rearrangement of Vector-Valued Functions
TL;DR: In this paper, it was shown that for every vector-valued function u Lp(X, p; Rd) there is a unique polar factorization u = V$.s, where $ is a convex function defined on R and s is a measure-preserving mapping from (x, p) into (Q, I. I), provided that u is nondegenerate.
Journal ArticleDOI
The geometry of dissipative evolution equations: the porous medium equation
TL;DR: In this paper, the authors show that the porous medium equation has a gradient flow structure which is both physically and mathematically natural, and they use the intuition and the calculus of Riemannian geometry to quantify this asymptotic behavior.
Book
Topological methods in hydrodynamics
Vladimir I. Arnold,Boris Khesin +1 more
TL;DR: A group theoretical approach to hydrodynamics is proposed in this article, where the authors consider the hydrodynamic geometry of diffeomorphism groups and the principle of least action implies that the motion of a fluid is described by geodesics on the group in the right-invariant Riemannian metric given by the kinetic energy.
Journal ArticleDOI
The variational formulation of the Fokker-Planck equation
TL;DR: The Fokker-Planck equation as mentioned in this paper describes the evolution of the probability density for a stochastic process associated with an Ito Stochastic Differential Equation.