Generalization of an Inequality by Talagrand and Links with the Logarithmic Sobolev Inequality
Felix Otto,Cédric Villani +1 more
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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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Dissertation
Statistical learning algorithms for geometric and topological data analysis
TL;DR: The concept of persistent homology, a concept of algebraic topology, is used to improve the pooling step of the bag-of-words approach for 3D shapes to improve its efficiency on both real and synthetic data.
Posted Content
Dimension-free Information Concentration via Exp-Concavity
Ya-Ping Hsieh,Volkan Cevher +1 more
TL;DR: In this article, it was shown that if the potentials of the log-concave distribution are exp-Concave, which is a central notion for fast rates in online and statistical learning, then the concentration of information can be further improved to depend only on the expconcavity parameter and hence, it can be dimension independent.
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A transportation approach to the mean-field approximation
TL;DR: These transportation-entropy inequalities which are saturated for measures such that their log-density with respect to the background measure is an affine function, in the setting of the uniform measure on the discrete hypercube and the exponential measure imply a strong integrability inequality for Bernoulli and exponential processes.
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Transportation-cost inequalities on path spaces over manifolds carrying geometric flows
TL;DR: In this article, the authors proved that the curvature lower bound condition of the Ricci flow with the umbilic boundary is equivalent to the convexity of the geometric flow (i.e., the boundary keeps convex).
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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.