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
Citations
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Optimal Transport and Ricci Curvature for Metric- Measure Spaces
TL;DR: We refer to as mentioned in this paper for a survey of lower Ricci curvature bounds for metric-measure spaces with respect to the Gromov-Hausdorff topology.
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R\'enyi divergence and the central limit theorem
TL;DR: In this paper, the authors explore properties of the R\'chi^2$ and more general R\'enyi (Tsallis) distances to the normal law and derive exact rates of convergence in these distances with respect to an increasing number of summands.
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Generalized No Free Lunch Theorem for Adversarial Robustness
TL;DR: It is shown that if conditioned on a class label the data distribution satisfies the Talagrand transportation-cost inequality, any classifier can be adversarially fooled with high probability once the perturbations are slightly greater than the natural noise level in the problem.
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Generative Modeling with Denoising Auto-Encoders and Langevin Sampling.
TL;DR: It is shown that both DAE and DSM provide estimates of the score of the Gaussian smoothed population density, allowing the machinery of Empirical Processes to apply to the homotopy method of arXiv:1907.05600.
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Non-Maxwellian kinetic equations modeling the dynamics of wealth distribution
TL;DR: In this paper, a class of new one-dimensional linear Fokker-Planck-type equations describing the dynamics of the distribution of wealth in a multi-agent society is introduced.
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.
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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.
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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.
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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.