Results 1  10
of
30
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
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Spectral Gap, Logarithmic Sobolev Constant, and Geometric Bunds
 Surveys in Diff. Geom., Vol. IX, 219–240, Int
, 2004
"... We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov ch ..."
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Cited by 14 (0 self)
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We survey recent works on the connection between spectral gap and logarithmic Sobolev constants, and exponential integrability of Lipschitz functions. In particular, tools from measure concentration are used to describe bounds on the diameter of a (compact) Riemannian manifold and of Markov chains in terms of the first eigenvalue of the Laplacian and the logarithmic Sobolev constant. We examine similarly dimension free isoperimetric bounds using these parameters.
Characterization of Talagrand’s like transportationcost inequalities on the real line
, 2006
"... In this paper, we give necessary and sufficient conditions for Talagrand’s like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimensionfree concentration of measure property. Another byproduct is the characterization ..."
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Cited by 13 (5 self)
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In this paper, we give necessary and sufficient conditions for Talagrand’s like transportation cost inequalities on the real line. This brings a new wide class of examples of probability measures enjoying a dimensionfree concentration of measure property. Another byproduct is the characterization of modified LogSobolev inequalities for Logconcave probability measures on R.
Semidefinite Relaxation Bounds for Indefinite Homogeneous Quadratic Optimization
, 2007
"... This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x ∗ Cx  x ∗ Akx ≥ 1, k = 0, 1,..., m, x ∈ F n} and (2) max{x ∗ Cx  x ∗ Akx ..."
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Cited by 11 (5 self)
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This paper studies the relationship between the optimal value of a homogeneous quadratic optimization problem and that of its Semidefinite Programming (SDP) relaxation. We consider two quadratic optimization models: (1) min{x ∗ Cx  x ∗ Akx ≥ 1, k = 0, 1,..., m, x ∈ F n} and (2) max{x ∗ Cx  x ∗ Akx ≤ 1, k = 0, 1,..., m, x ∈ F n}, where F is either the real field R or the complex field C, and Ak, C are symmetric matrices. For the minimization model (1), we prove that, if the matrix C and all but one of Ak’s are positive semidefinite, then the ratio between the optimal value of (1) and its SDP relaxation is upper bounded by O(m 2) when F = R, and by O(m) when F = C. Moreover, when two or more of Ak’s are indefinite, this ratio can be arbitrarily large. For the maximization model (2), we show that, if C and at most one of Ak’s are indefinite while other Ak’s are positive semidefinite, then the ratio between the optimal value of (2) and its SDP relaxation is bounded from below by O(1 / log m) for both the real and complex case. This result improves the bound based on the socalled approximate SLemma of BenTal et al. [3]. When two or more of Ak in (2) are indefinite, we derive a general bound in terms of the problem data and the SDP solution. For both optimization models, we present examples to show that the derived
The subgaussian constant and concentration inequalities
"... We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to the spread constant, introduced by Alon, Boppana, and Spencer. We also settle, in the affirmative, ..."
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Cited by 9 (2 self)
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We study concentration inequalities for Lipschitz functions on graphs by estimating the optimal constant in exponential moments of subgaussian type. This is illustrated on various graphs and related to the spread constant, introduced by Alon, Boppana, and Spencer. We also settle, in the affirmative, a question of Talagrand on a deviation inequality for the discrete cube.
A CHARACTERIZATION OF DIMENSION FREE CONCENTRATION IN TERMS OF TRANSPORTATION INEQUALITIES
, 2008
"... The aim of this paper is to show that a probability measure µ on R d concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand’s T2 transportationcost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villan ..."
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Cited by 7 (4 self)
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The aim of this paper is to show that a probability measure µ on R d concentrates independently of the dimension like a gaussian measure if and only if it verifies Talagrand’s T2 transportationcost inequality. This theorem permits us to give a new and very short proof of a result of Otto and Villani. Generalizations to other types of concentration are also considered. In particular, one shows that the Poincaré inequality is equivalent to a certain form of dimension free exponential concentration. The proofs of these results rely on simple Large Deviations techniques.
TRANSPORT INEQUALITIES. A SURVEY
"... Abstract. This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory. ..."
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Cited by 7 (1 self)
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Abstract. This is a survey of recent developments in the area of transport inequalities. We investigate their consequences in terms of concentration and deviation inequalities and sketch their links with other functional inequalities and also large deviation theory.
Transportation Approach to Some Concentration Inequalities in Product Spaces
, 1996
"... Using a transportation approach we prove that for every probability measures P; Q 1 ; Q 2 on\Omega N with P a product measure there exist r.c.p.d. j such that R j (\Deltajx)dP (x) = Q j (\Delta) and Z dP (x) Z dP dQ 1 (y) fi dP dQ 2 (z) fi (1 + fi(1 \Gamma 2fi)) fN (x;y;z) d 1 (yjx ..."
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Cited by 6 (1 self)
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Using a transportation approach we prove that for every probability measures P; Q 1 ; Q 2 on\Omega N with P a product measure there exist r.c.p.d. j such that R j (\Deltajx)dP (x) = Q j (\Delta) and Z dP (x) Z dP dQ 1 (y) fi dP dQ 2 (z) fi (1 + fi(1 \Gamma 2fi)) fN (x;y;z) d 1 (yjx)d 2 (zjx) 1 ; for every fi 2 (0; 1=2). Here f N counts the number of coordinates k for which x k 6= y k and x k 6= z k . In case Q 1 = Q 2 one may take 1 = 2 . In the special case of Q j (\Delta) = P (\DeltajA) we recover some of Talagrand's sharper concentration inequalities in product spaces. In [Tal95, Tal96a], Talagrand provides a variety of concentration of measure inequalities which apply in every product space\Omega N (with\Omega Polish) equipped with a Borel product (probability) measure P . These inequalities are extremely useful in combinatorial applications such as the longest common/increasing subsequence, in statistical physics applications such as the study of spin glass...
Wave equations for graphs and the edgebased Laplacian, in "Pacific
 Journal of Mathematics
, 2004
"... In this paper we develop a wave equation for graphs that has many of the properties of the classical Laplacian wave equation. This wave equation is based on a type of graph Laplacian we call the “edgebased ” Laplacian. We give some applications of this wave equation to eigenvalue/geometry inequalit ..."
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Cited by 6 (0 self)
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In this paper we develop a wave equation for graphs that has many of the properties of the classical Laplacian wave equation. This wave equation is based on a type of graph Laplacian we call the “edgebased ” Laplacian. We give some applications of this wave equation to eigenvalue/geometry inequalities on graphs. 1
Concentration for infinitely divisible vectors with independent components. Available at http://arXiv.org/abs/math.PR/0606752 [RB] [R] [T
, 2004
"... For various classes of Lipschitz functions we provide dimension free concentration inequalities for infinitely divisible random vectors with independent components and finite exponential moments. The purpose of this note is to further visit the concentration phenomenon for infinitely divisible vecto ..."
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Cited by 5 (2 self)
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For various classes of Lipschitz functions we provide dimension free concentration inequalities for infinitely divisible random vectors with independent components and finite exponential moments. The purpose of this note is to further visit the concentration phenomenon for infinitely divisible vectors with independent components in an attempt to obtain dimension free concentration. Let X ∼ ID(γ, 0, ν) be an infinitely divisible (i.d.) vector (without Gaussian component) in R d, and with characteristic function ϕ(t) = Ee i〈t,X 〉 , t ∈ R d (throughout, 〈·, · 〉 denotes the Euclidean inner product in R d, while ‖ · ‖ is the corresponding Euclidean norm). As well known, ϕ(t) = exp i〈t, γ 〉 + Rd (e i〈t,u 〉} − 1 − i〈t, u〉1‖u‖≤1)ν(du) , (1) where γ ∈ R d and where ν ̸ ≡ 0 (the Lévy measure) is a positive Borel measure on R d, without atom at the origin and such that ∫ R d(1 ∧ ‖u ‖ 2)ν(du) < +∞. As also well known, X has independent components if and only if ν is supported on the axes of R d, i.e., ν(dx1,...,dxd) = d∑ δ0(dx1) · · ·δ0(dxk−1)˜νk(dxk)δ0(dxk+1) · · ·δ0(dxd). (2) k=1