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15
Exponential integrability and transportation cost related to logarithmic Sobolev inequalities
 J. Funct. Anal
, 1999
"... We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarith ..."
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Cited by 80 (4 self)
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We study some problems on exponential integrability, concentration of measure, and transportation cost related to logarithmic Sobolev inequalities. On the real line, we then give a characterization of those probability measures which satisfy these inequalities 1999 Academic Press Key Words: logarithmic Sobolev inequalities; exponential integrability; concentration of measure; transportation inequalities.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
The entropy formula for linear heat equation
 J. Geom. Anal
, 2004
"... ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp ..."
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Cited by 28 (11 self)
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ABSTRACT. We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the wellknown Li–Yau’s gradient estimate. As a byproduct we obtain the sharp estimates on ‘Nash’s entropy ’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li–Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to R n. In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman’s entropy in the case of Riemann surfaces. 1. The relation with Li–Yau’s gradient estimates In this section we provide another derivation of Theorem 1.1 of [8] and discuss its relation with Li–Yau’s gradient estimates on positive solutions of heat equation. The formulation gives a sharp upper and lower bound estimates on Nash’s ‘entropy quantity ’ − � M H log Hdvin the case M has nonnegative Ricci curvature, where H is the fundamental solution (heat kernel) of the heat equation. This section is following the ideas in the Section 5 of [9]. Let u(x, t) be a positive solution to � ∂ ∂t − � � u(x, t) = 0 with �
Quasifactorization of the entropy and logarithmic Sobolev inequalities for Gibbs random fields
, 2001
"... . We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, ..."
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Cited by 21 (0 self)
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. We show that the entropy functional exhibits a quasi{factorization property with respect to a pair of weakly dependent {algebras. As an application we give a simple proof that the Dobrushin and Shlosman's complete analyticity condition, for a Gibbs specication with nite range summable interaction, implies uniform logarithmic Sobolev inequalities. This result has been previously proven using several dierent techniques. The advantage of our approach is that it relies almost entirely on a general property of the entropy, while very little is assumed on the Dirichlet form. No topology is introduced on the single spin space, thus discrete and continuous spins can be treated in the same way. Key Words: Entropy, Logarithmic Sobolev inequalities, Gibbs measures Mathematics Subject Classication 2000: 82B20, 82C20, 39B62 v1.0 1. Introduction Logarithmic Sobolev inequalities have been introduced in [Gr1] where it has been shown that Z R d f 2 (x) log jf(x)j d (dx) Z R d jr...
Isoperimetry between exponential and Gaussian
 Electronic J. Prob
"... We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1 ..."
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Cited by 16 (7 self)
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We study in details the isoperimetric profile of product probability measures with tails between the exponential and the Gaussian regime. In particular we exhibit many examples where coordinate halfspaces are approximate solutions of the isoperimetric problem. 1
Weighted Poincarétype inequalities for Cauchy and other convex measures
 Annals of Probability
, 2007
"... Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the ..."
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Cited by 9 (1 self)
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Brascamp–Liebtype, weighted Poincarétype and related analytic inequalities are studied for multidimensional Cauchy distributions and more general κconcave probability measures (in the hierarchy of convex measures). In analogy with the limiting (infinitedimensional logconcave) Gaussian model, the weighted inequalities fully describe the measure concentration and large deviation properties of this family of measures. Cheegertype isoperimetric inequalities are investigated similarly, giving rise to a common weight in the class of concave probability measures under consideration. 1. Introduction. The
Beyond Spectral Clustering  Tight Relaxations of Balanced Graph Cuts
 In In Advances in Neural Information Processing Systems (NIPS
"... Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a nonlinear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result consid ..."
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Cited by 6 (3 self)
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Spectral clustering is based on the spectral relaxation of the normalized/ratio graph cut criterion. While the spectral relaxation is known to be loose, it has been shown recently that a nonlinear eigenproblem yields a tight relaxation of the Cheeger cut. In this paper, we extend this result considerably by providing a characterization of all balanced graph cuts which allow for a tight relaxation. Although the resulting optimization problems are nonconvex and nonsmooth, we provide an efficient firstorder scheme which scales to large graphs. Moreover, our approach comes with the quality guarantee that given any partition as initialization the algorithm either outputs a better partition or it stops immediately. 1
Some Connections between Sobolevtype Inequalities and Isoperimetry
 Memoirs of the Amer. Math. Soc
, 1995
"... Introduction Motivation, Examples, Statements of Results It is well known that the Sobolev inequality Z R n jrf(x)jd¯(x) n! 1=n n `Z R n jf(x)j n n\Gamma1 d¯(x) 'n\Gamma1 n ; n 2; (1.1) where f is a compactly supported smooth function on the Euclidean space R n , where ¯ is the L ..."
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Cited by 3 (2 self)
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Introduction Motivation, Examples, Statements of Results It is well known that the Sobolev inequality Z R n jrf(x)jd¯(x) n! 1=n n `Z R n jf(x)j n n\Gamma1 d¯(x) 'n\Gamma1 n ; n 2; (1.1) where f is a compactly supported smooth function on the Euclidean space R n , where ¯ is the Lebesgue measure in R n , and where ! n is the volume of the unit ball in R n , is equivalent to the isoperimetric property of the balls in R n . This isoperimetric property
LOGARITHMIC SOBOLEV INEQUALITIES FOR INFINITE DIMENSIONAL HÖRMANDER TYPE GENERATORS ON THE HEISENBERG GROUP
, 901
"... Abstract. The Heisenberg group is one of the simplest subRiemannian settings in which we can define nonelliptic Hörmander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of nontrivial Gibbs measures with quadratic interaction ..."
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Cited by 2 (1 self)
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Abstract. The Heisenberg group is one of the simplest subRiemannian settings in which we can define nonelliptic Hörmander type generators. We can then consider coercive inequalities associated to such generators. We prove that a certain class of nontrivial Gibbs measures with quadratic interaction potential on an infinite product of Heisenberg groups satisfy logarithmic Sobolev inequalities. 1.
L qfunctional inequalities and weighted porous media equations
, 2008
"... Using measurecapacity inequalities we study new functional inequalities, namely LqPoincaré inequalities Varµ(f q) 1/q ∫ ..."
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Using measurecapacity inequalities we study new functional inequalities, namely LqPoincaré inequalities Varµ(f q) 1/q ∫