Results 1  10
of
43
Concentration inequalities
 ADVANCED LECTURES IN MACHINE LEARNING
, 2004
"... Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis o ..."
Abstract

Cited by 62 (1 self)
 Add to MetaCart
(Show Context)
Concentration inequalities deal with deviations of functions of independent random variables from their expectation. In the last decade new tools have been introduced making it possible to establish simple and powerful inequalities. These inequalities are at the heart of the mathematical analysis of various problems in machine learning and made it possible to derive new efficient algorithms. This text attempts to summarize some of the basic tools.
Interpolated inequalities between exponential and Gaussian, Orlicz hypercontractivity and isoperimetry
, 2004
"... ..."
(Show Context)
On the role of convexity in isoperimetry, spectralgap and concentration
 Invent. Math
"... We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitativ ..."
Abstract

Cited by 19 (5 self)
 Add to MetaCart
(Show Context)
We show that for convex domains in Euclidean space, Cheeger’s isoperimetric inequality, spectral gap of the Neumann Laplacian, exponential concentration of Lipschitz functions, and the apriori weakest requirement that Lipschitz functions have arbitrarily slow uniform taildecay, are all quantitatively equivalent (to within universal constants, independent of the dimension). This substantially extends previous results of Maz’ya, Cheeger, Gromov– Milman, Buser and Ledoux. As an application, we conclude a sharp quantitative stability result for the spectral gap of convex domains under convex perturbations which preserve volume (up to constants) and under maps which are “onaverage ” Lipschitz. We also provide a new characterization (up to constants) of the spectral gap of a convex domain, as one over the square of the average distance from the “worst ” subset having half the measure of the domain. In addition, we easily recover and extend many previously known lower bounds on the spectral gap of convex domains, due to Payne–Weinberger, Li–Yau, Kannan– Lovász–Simonovits, Bobkov and Sodin. The proof involves estimates on the diffusion semigroup following Bakry–Ledoux and a result from Riemannian Geometry on the concavity of the isoperimetric profile. Our results extend to the more general setting of Riemannian manifolds with density which satisfy the CD(0, ∞) curvaturedimension condition of BakryÉmery. 1
INTERPOLATION BETWEEN LOGARITHMIC SOBOLEV AND POINCARÉ INEQUALITIES
"... Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 1. ..."
Abstract

Cited by 17 (7 self)
 Add to MetaCart
(Show Context)
Abstract. This note is concerned with intermediate inequalities which interpolate between the logarithmic Sobolev and the Poincaré inequalities. For such generalized Poincaré inequalities we improve upon the known constants from the literature. 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. ..."
Abstract

Cited by 17 (8 self)
 Add to MetaCart
(Show Context)
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.
BinomialPoisson entropic inequalities and the M/M/∞ queue
 ESAIM Probab. Stat
"... This article provides entropic inequalities for binomialPoisson distributions, derived from the two point space. They appear as local inequalities of the M/M/ ∞ queue. They describe in particular the exponential dissipation of Φentropies along this process. This simple queueing process appears as ..."
Abstract

Cited by 17 (5 self)
 Add to MetaCart
(Show Context)
This article provides entropic inequalities for binomialPoisson distributions, derived from the two point space. They appear as local inequalities of the M/M/ ∞ queue. They describe in particular the exponential dissipation of Φentropies along this process. This simple queueing process appears as a model of “constant curvature”, and plays for the simple Poisson process the role played by the OrnsteinUhlenbeck process for Brownian Motion. Some of the inequalities are recovered by semigroup interpolation. Additionally, we explore the behaviour of these entropic inequalities under a particular scaling, which sees the OrnsteinUhlenbeck process as a fluid limit of M/M/ ∞ queues. Proofs are elementary and rely essentially on the development of a “Φcalculus”.
Concentration for independent random variables with heavy tails
 AMRX
, 2005
"... If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 1 ..."
Abstract

Cited by 15 (9 self)
 Add to MetaCart
(Show Context)
If a random variable is not exponentially integrable, it is known that no concentration inequality holds for an infinite sequence of independent copies. Under mild conditions, we establish concentration inequalities for finite sequences of n independent copies, with good dependence in n. 1
Refined convex Sobolev Inequalities
 J. Functional Analysis
, 2005
"... This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy–entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré type inequalities with weights are derived. Best const ..."
Abstract

Cited by 14 (6 self)
 Add to MetaCart
(Show Context)
This paper is devoted to refinements of convex Sobolev inequalities in the case of power law relative entropies: a nonlinear entropy–entropy production relation improves the known inequalities of this type. The corresponding generalized Poincaré type inequalities with weights are derived. Best constants are compared to the usual Poincaré constant. Key words and phrases: Sobolev inequality – Poincaré inequality – entropy method
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 ..."
Abstract

Cited by 13 (8 self)
 Add to MetaCart
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.
Mixing time bounds via the spectral profile
 Electron. J. Probab
, 2006
"... E l e c t r o n ..."
(Show Context)