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48
Gaussian processes: inequalities, small ball probabilities and applications
 STOCHASTIC PROCESSES: THEORY AND METHODS
, 2001
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Theory of classification: A survey of some recent advances
, 2005
"... The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results. ..."
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Cited by 93 (3 self)
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The last few years have witnessed important new developments in the theory and practice of pattern classification. We intend to survey some of the main new ideas that have led to these recent results.
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 ..."
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Cited by 89 (1 self)
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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.
Concentration inequalities using the entropy method
"... We investigate a new methodology, worked out by Ledoux and Massart, to prove concentrationofmeasure inequalities. The method is based on certain modified logarithmic Sobolev inequalities. We provide some very simple and general readytouse inequalities. One of these inequalities may be considered ..."
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Cited by 67 (3 self)
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We investigate a new methodology, worked out by Ledoux and Massart, to prove concentrationofmeasure inequalities. The method is based on certain modified logarithmic Sobolev inequalities. We provide some very simple and general readytouse inequalities. One of these inequalities may be considered as an exponential version of the EfronStein inequality. The main purpose of this paper is to point out the simplicity and the generality of the approach. We show how the new method can recover many of Talagrand’s revolutionary inequalities and provide new applications in a variety of problems including Rademacher averages, Rademacher chaos, the number of certain small subgraphs in a random graph, and the minimum of the empirical risk in some statistical estimation problems.
Approximation, Metric Entropy and Small Ball Estimates for Gaussian Measures
 Ann. Probab
, 1999
"... A precise link proved by J. Kuelbs and W. V. Li relates the small ball behavior of a Gaussian measure on a Banach space E with the metric entropy behavior of K , the unit ball of the RKHS of in E. We remove the main regularity assumption imposed on the unknown function in the link. This enables t ..."
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Cited by 63 (27 self)
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A precise link proved by J. Kuelbs and W. V. Li relates the small ball behavior of a Gaussian measure on a Banach space E with the metric entropy behavior of K , the unit ball of the RKHS of in E. We remove the main regularity assumption imposed on the unknown function in the link. This enables the application of tools and results from functional analysis to small ball problems and leads to small ball estimates of general algebraic type as well as to new estimates for concrete Gaussian processes. Moreover, we show that the small ball behavior of a Gaussian process is also tightly connected with the speed of approximation by "nite rank" processes. Abbreviated title: Metric Entropy and Small Ball Estimates Keywords: Gaussian process, small deviation, metric entropy, approximation number. AMS 1991 Subject Classications: Primary: 60G15 ; Secondary: 60F99, 47D50, 47G10 . 1 Supported in part by NSF 1 1 Introduction Let denote a centered Gaussian measure on a real separable B...
Differential equations driven by Gaussian signals
, 2007
"... We consider multidimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful ye ..."
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Cited by 59 (15 self)
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We consider multidimensional Gaussian processes and give a new condition on the covariance, simple and sharp, for the existence of Lévy area(s). Gaussian rough paths are constructed with a variety of weak and strong approximation results. Together with a new RKHS embedding, we obtain a powerful yet conceptually simple framework in which to analysize differential equations driven by Gaussian signals in the rough paths sense. 1
Sample path properties of anisotropic Gaussian random fields
, 2008
"... Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operatorscaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path propert ..."
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Cited by 45 (16 self)
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Anisotropic Gaussian random fields arise in probability theory and in various applications. Typical examples are fractional Brownian sheets, operatorscaling Gaussian fields with stationary increments, and the solution to the stochastic heat equation. This paper is concerned with sample path properties of anisotropic Gaussian random fields in general. Let X = {X(t), t ∈ RN} be a Gaussian random field with values in Rd and with parameters H1,..., HN. Our goal is to characterize the anisotropic nature of X in terms of its parameters explicitly. Under some general conditions, we establish results on the modulus of continuity, small ball probabilities, fractal dimensions, hitting probabilities and local times of anisotropic Gaussian random fields. An important tool for our study is the various forms of strong local nondeterminism.
Concentration and Deviation Inequalities in Infinite Dimensions via Covariance Representations
, 2002
"... Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative ..."
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Cited by 35 (12 self)
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Concentration and deviation inequalities are obtained for functionals on Wiener space, Poisson space or more generally for normal martingales and binomial processes. The method used here is based on covariance identities obtained via the chaotic representation property, and provides an alternative to the use of logarithmic Sobolev inequalities. It allows to recover known concentration and deviation inequalities on the Wiener and Poisson space (including the ones given by sharp logarithmic Sobolev inequalities), and extends results available in the discrete case, i.e. on the infinite cube {1; 1}∞.
Large deviations and isoperimetry over convex probability measures with heavy tails
 Electron J. Prob
, 2007
"... Large deviations and isoperimetric inequalities are considered for probability distributions,
satisfying convexity conditions of the BrunnMinkowskitype. ..."
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Cited by 27 (4 self)
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Large deviations and isoperimetric inequalities are considered for probability distributions,
satisfying convexity conditions of the BrunnMinkowskitype.