Results 1  10
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24
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. ..."
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Cited by 175 (9 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.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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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 38 (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 �
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, th ..."
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Cited by 30 (3 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.
Mass transport and variants of the logarithmic Sobolev inequality
 J. Geometric Analysis
"... Abstract. We develop the optimal transportation approach to modified logSobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical logSobolev case. The idea behind many of these conditions is tha ..."
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Cited by 28 (0 self)
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Abstract. We develop the optimal transportation approach to modified logSobolev inequalities and to isoperimetric inequalities. Various sufficient conditions for such inequalities are given. Some of them are new even in the classical logSobolev case. The idea behind many of these conditions is that measures with a nonconvex potential may enjoy such functional inequalities provided they have a strong integrability property that balances the lack of convexity. In addition, several known criteria are recovered in a simple unified way by transportation methods and generalized to the Riemannian setting.
Quasifactorization of the entropy and logarithmic sobolev inequalities for gibbs random elds
, 2001
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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 24 (7 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
Isoperimetry between exponential and Gaussian
 ELECTRONIC J. PROB
, 2008
"... 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. ..."
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Cited by 23 (8 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.
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 10 (5 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.
Variational formulas of Poincarétype inequalities in Banach spaces of functions on
 the line, Acta
, 2002
"... Abstract. In an author’s previous paper, the same topic was studied for one dimensional diffusions. As a continuation, this paper studies the discrete case, that is the birthdeath processes. The explicit criteria for the inequalities, the variational formulas and explicit bounds of the correspondin ..."
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Cited by 8 (4 self)
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Abstract. In an author’s previous paper, the same topic was studied for one dimensional diffusions. As a continuation, this paper studies the discrete case, that is the birthdeath processes. The explicit criteria for the inequalities, the variational formulas and explicit bounds of the corresponding constants in the inequalities are presented. As typical applications, the Nash inequalities and logarithmic Sobolev inequalities are examined. This paper is a continuation of [1] to deal with the discrete case.