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Muli Safra - One of the best experts on this subject based on the ideXlab platform.
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theorems of kkl friedgut and Talagrand via random restrictions and log sobolev inequality
Conference on Innovations in Theoretical Computer Science, 2021Co-Authors: Esty Kelman, Subhash Khot, Dor Minzer, Guy Kindler, Muli SafraAbstract:We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.
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on monotonicity testing and boolean isoperimetric type theorems
SIAM Journal on Computing, 2018Co-Authors: Subhash Khot, Dor Minzer, Muli SafraAbstract:We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm...
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on monotonicity testing and boolean isoperimetric type theorems
Foundations of Computer Science, 2015Co-Authors: Subhash Khot, Dor Minzer, Muli SafraAbstract:We show a directed and robust analogue of a boolean isoperimetric type theorem of Talagrand. As an application, we give a monotonicity testing algorithm that makes, up to polylog factors, (square root of n)/a#x03B5;2 non-adaptive queries to a function f:0, 1n -> 0, 1, always accepts a monotone function and rejects a function that is a#x03B5;-far from being monotone with constant probability.
Prasad Tetali - One of the best experts on this subject based on the ideXlab platform.
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KANTOROVICH DUALITY FOR GENERAL TRANSPORT COSTS AND APPLICATIONS
Journal of Functional Analysis, 2018Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie Samson, Prasad TetaliAbstract:We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.
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convergence to global equilibrium for fokker planck equations on a graph and Talagrand type inequalities
Journal of Differential Equations, 2016Co-Authors: Wen Huang, Yao Li, Prasad TetaliAbstract:Abstract In 2012, Chow, Huang, Li and Zhou [7] proposed the Fokker–Planck equations for the free energy on a finite graph, in which they showed that the corresponding Fokker–Planck equation is a nonlinear ODE defined on a Riemannian manifold of probability distributions. Different choices for inner products result in different Fokker–Planck equations. The unique global equilibrium of each equation is a Gibbs distribution. In this paper we proved that the exponential rate of convergence towards the global equilibrium of these Fokker–Planck equations. The rate is measured by both the decay of the L 2 norm and that of the (relative) entropy. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [7] . The first one is a local inequality, while the second is a global inequality with respect to the “lower bound metric” from [7] .
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convergence to global equilibrium for fokker planck equations on a graph and Talagrand type inequalities
arXiv: Classical Analysis and ODEs, 2014Co-Authors: Wen Huang, Yao Li, Prasad TetaliAbstract:In recent work, Chow, Huang, Li and Zhou introduced the study of Fokker-Planck equations for a free energy function defined on a finite graph. When $N\ge 2$ is the number of vertices of the graph, they show that the corresponding Fokker-Planck equation is a system of $N$ nonlinear ordinary differential equations defined on a Riemannian manifold of probability distributions. The different choices for inner products on the space of probability distributions result in different Fokker-Planck equations for the same process. Each of these Fokker-Planck equations has a unique global equilibrium, which is a Gibbs distribution. In this paper we study the {\em speed of convergence} towards global equilibrium for the solution of these Fokker-Planck equations on a graph, and prove that the convergence is indeed exponential. The rate as measured by the decay of the $L_2$ norm can be bound in terms of the spectral gap of the Laplacian of the graph, and as measured by the decay of (relative) entropy be bound using the modified logarithmic Sobolev constant of the graph. With the convergence result, we also prove two Talagrand-type inequalities relating relative entropy and Wasserstein metric, based on two different metrics introduced in [CHLZ] The first one is a local inequality, while the second is a global inequality with respect to the "lower bound metric" from [CHLZ].
Cyril Roberto - One of the best experts on this subject based on the ideXlab platform.
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KANTOROVICH DUALITY FOR GENERAL TRANSPORT COSTS AND APPLICATIONS
Journal of Functional Analysis, 2018Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie Samson, Prasad TetaliAbstract:We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.
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some remarks on the stability of the log sobolev inequality for the gaussian measure
Potential Analysis, 2017Co-Authors: Emanuel Indrei, Maria Rosaria Posteraro, Cyril RobertoAbstract:This note consists of two parts. Firstly, we bound the deficit in the logarithmic Sobolev Inequality and in the Talagrand transport-entropy Inequality for the Gaussian measure, in any dimension, by means of a distance introduced by Bucur and Fragala. Thereafter, we investigate the stability issue with tools from Fourier analysis.
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DEVIATION INEQUALITIES FOR CONVEX FUNCTIONS MOTIVATED BY THE Talagrand CONJECTURE
2017Co-Authors: Nathael Gozlan, Cyril Roberto, M Madiman, Paul-marie SamsonAbstract:Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.
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A new characterization of Talagrand’s transportentropy inequalities and applications
2016Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie Samson, Université Paris-est Marne-la-valléeAbstract:We show that Talagrand’s transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand’s inequality is stable under bounded pertur-bations. 1. Introduction. Talagrand’
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characterization of Talagrand s transport entropy inequalities in metric spaces
Annals of Probability, 2013Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie SamsonAbstract:We give a characterization of transport-entropy inequalities in metric spaces. As an application we deduce that such inequalities are stable under bounded perturbation (Holley–Stroock perturbation lemma).
Paul-marie Samson - One of the best experts on this subject based on the ideXlab platform.
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KANTOROVICH DUALITY FOR GENERAL TRANSPORT COSTS AND APPLICATIONS
Journal of Functional Analysis, 2018Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie Samson, Prasad TetaliAbstract:We introduce a general notion of transport cost that encompasses many costs used in the literature (including the classical one and weak transport costs introduced by Talagrand and Marton in the 90's), and prove a Kantorovich type duality theorem. As a by-product we obtain various applications in different directions: we give a short proof of a result by Strassen on the existence of a martingale with given marginals, we characterize the associated transport-entropy inequalities together with the log-Sobolev inequality restricted to convex/concave functions. Some explicit examples of discrete measures satisfying weak transport-entropy inequalities are also given.
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DEVIATION INEQUALITIES FOR CONVEX FUNCTIONS MOTIVATED BY THE Talagrand CONJECTURE
2017Co-Authors: Nathael Gozlan, Cyril Roberto, M Madiman, Paul-marie SamsonAbstract:Motivated by Talagrand's conjecture on regularization properties of the natural semigroup on the Boolean hypercube, and in particular its continuous analogue involving regularization properties of the Ornstein-Uhlenbeck semigroup acting on in-tegrable functions, we explore deviation inequalities for log-semiconvex functions under Gaussian measure.
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A new characterization of Talagrand’s transportentropy inequalities and applications
2016Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie Samson, Université Paris-est Marne-la-valléeAbstract:We show that Talagrand’s transport inequality is equivalent to a restricted logarithmic Sobolev inequality. This result clarifies the links between these two important functional inequalities. As an application, we give the first proof of the fact that Talagrand’s inequality is stable under bounded pertur-bations. 1. Introduction. Talagrand’
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characterization of Talagrand s transport entropy inequalities in metric spaces
Annals of Probability, 2013Co-Authors: Nathael Gozlan, Cyril Roberto, Paul-marie SamsonAbstract:We give a characterization of transport-entropy inequalities in metric spaces. As an application we deduce that such inequalities are stable under bounded perturbation (Holley–Stroock perturbation lemma).
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Infimum-convolution description of concentration properties of product probability measures, with applications
Annales de l'Institut Henri Poincaré (B) Probabilités et Statistiques, 2007Co-Authors: Paul-marie SamsonAbstract:This paper is devoted to the concentration properties of product probability measures mu = mu(1) circle times(...)circle times mu(n), expressed in term of dimension-free functional inequalities of the form [GRAPHICS] where a is a parameter, 0 < alpha < 1, and Q(alpha)f is an appropriate infimum-convolution operator. This point of view has been introduced by Maurey [B. Maurey, Some deviation inequalities, Geom. Funct: Anal. 1 (1991) 188-197]. It has its origins in concentration inequalities by Talagrand where the enlargement of sets is done in accordance with the cost function of the operator Q alpha f (see [M. Talagrand, Concentration of measure and isoperimetric, inequalities in product spaces, Publ. Math. Inst. Hautes Etudes Sci. 81 (1995) 73-205, M. Talagrand, New concentration inequalities in product spaces, Invent. Math. 126 *(1996) 505-563, M. Talagrand, A new look at independence, Ann. Probab. 24 (1996) 1-34]). A main application of the functional inequalities obtained here is optimal deviations inequalities for suprema of sums of independent random variables. As example, we also derive classical deviations bounds for the one-dimensional bin packing problem. (c) 2006 Elsevier Masson SAS. All rights reserved.
Subhash Khot - One of the best experts on this subject based on the ideXlab platform.
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theorems of kkl friedgut and Talagrand via random restrictions and log sobolev inequality
Conference on Innovations in Theoretical Computer Science, 2021Co-Authors: Esty Kelman, Subhash Khot, Dor Minzer, Guy Kindler, Muli SafraAbstract:We give alternate proofs for three related results in analysis of Boolean functions, namely the KKL Theorem, Friedgut’s Junta Theorem, and Talagrand’s strengthening of the KKL Theorem. We follow a new approach: looking at the first Fourier level of the function after a suitable random restriction and applying the Log-Sobolev inequality appropriately. In particular, we avoid using the hypercontractive inequality that is common to the original proofs. Our proofs might serve as an alternate, uniform exposition to these theorems and the techniques might benefit further research.
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on monotonicity testing and boolean isoperimetric type theorems
SIAM Journal on Computing, 2018Co-Authors: Subhash Khot, Dor Minzer, Muli SafraAbstract:We show a directed and robust analogue of a boolean isoperimetric-type theorem of Talagrand [Geom. Funct. Anal., 3 (1993), pp. 295--314]. As an application, we give a monotonicity testing algorithm...
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on monotonicity testing and boolean isoperimetric type theorems
Foundations of Computer Science, 2015Co-Authors: Subhash Khot, Dor Minzer, Muli SafraAbstract:We show a directed and robust analogue of a boolean isoperimetric type theorem of Talagrand. As an application, we give a monotonicity testing algorithm that makes, up to polylog factors, (square root of n)/a#x03B5;2 non-adaptive queries to a function f:0, 1n -> 0, 1, always accepts a monotone function and rejects a function that is a#x03B5;-far from being monotone with constant probability.