Upper and Lower Bounds for Stochastic Processes

Talagrand, Michel

doi:10.1007/978-3-642-54075-2

Cited by 458 publications

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“…standard Gaussian variables is of order log(j + 1) [7,Exercise 2.2.7]. It remains to take the maximum over j ≤ n.…”

Section: Gaussian Estimatesmentioning

confidence: 99%

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On the spectral norm of Gaussian random matrices

Handel

2017

Trans. Amer. Math. Soc.

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Abstract. Let X be a d × d symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata la that the spectral norm of X is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous Gaussian random matrices. This paper establishes the conjecture up to a dimensional factor of order √ log log d. Moreover, dimensionfree bounds are developed that are optimal to leading order and that establish the conjecture in special cases. The proofs of these results shed significant light on the geometry of the underlying Gaussian processes.

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“…standard Gaussian variables is of order log(j + 1) [7,Exercise 2.2.7]. It remains to take the maximum over j ≤ n.…”

Section: Gaussian Estimatesmentioning

confidence: 99%

“…In general, the connection between the supremum of a Gaussian process and the underlying geometry is described by the generic chaining method [7]. Unfortunately, even a geometric description along these lines of the trivial behavior of the supremum of a Gaussian process over a convex hull remains a long-standing open problem [7, pp.…”

Section: 3mentioning

confidence: 99%

On the spectral norm of Gaussian random matrices

Handel

2017

Trans. Amer. Math. Soc.

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“…In the study of the supremum of a stochastic process indexed by a set S, it is natural to endow the set with a pseudo-metric d. The structure and "richness" of the index set S (as given by covering numbers or, more generally, via the chaining technique [31,33]) yield precise control on the supremum of the stochastic process. It is natural to ask whether we can endow the projection F(z) with a metric and appeal to known results.…”

Section: Definitionmentioning

confidence: 99%

Sequential complexities and uniform martingale laws of large numbers

Rakhlin

Sridharan

Tewari

2014

Probab. Theory Relat. Fields

122

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We establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide "sequential" (noni.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform convergence rates for empirical processes with dependent data. As a direct application of our results, we provide exponential inequalities for sums of martingale differences in Banach spaces.Keywords empirical processes, dependent data, uniform Glivenko-Cantelli classes, rademacher averages, sequential prediction Disciplines Statistics and ProbabilityThis journal article is available at ScholarlyCommons: http://repository.upenn.edu/statistics_papers/531 Sequential Complexities and Uniform Martingale Laws of Large NumbersAlexander Rakhlin Karthik Sridharan Ambuj TewariSeptember 19, 2013 AbstractWe establish necessary and sufficient conditions for a uniform martingale Law of Large Numbers. We extend the technique of symmetrization to the case of dependent random variables and provide "sequential" (non-i.i.d.) analogues of various classical measures of complexity, such as covering numbers and combinatorial dimensions from empirical process theory. We establish relationships between these various sequential complexity measures and show that they provide a tight control on the uniform convergence rates for empirical processes with dependent data. As a direct application of our results, we provide exponential inequalities for sums of martingale differences in Banach spaces.

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“…Powerful methods to estimate the mean width w(K) have been developed in connection with stochastic processses. These methods include Sudakov's and Dudley's inequalities which relate w(K) to the covering numbers of K in the Euclidean metric, and the sharp technique of majorizing measures (see [16,21]). …”

Section: 3mentioning

confidence: 99%

Dimension Reduction by Random Hyperplane Tessellations

Plan

Vershynin

2013

Discrete Comput Geom

100

141

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Abstract. Given a subset K of the unit Euclidean sphere, we estimate the minimal number m = m(K) of hyperplanes that generate a uniform tessellation of K, in the sense that the fraction of the hyperplanes separating any pair x, y ∈ K is nearly proportional to the Euclidean distance between x and y. Random hyperplanes prove to be almost ideal for this problem; they achieve the almost optimal bound m = O(w(K) 2 ) where w(K) is the Gaussian mean width of K. Using the map that sends x ∈ K to the sign vector with respect to the hyperplanes, we conclude that every bounded subset K of R n embeds into the Hamming cube {−1, 1} m with a small distortion in the Gromov-Haussdorff metric. Since for many sets K one has m = m(K) n, this yields a new discrete mechanism of dimension reduction for sets in Euclidean spaces.

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Upper and Lower Bounds for Stochastic Processes

Cited by 458 publications

References 0 publications

On the spectral norm of Gaussian random matrices

On the spectral norm of Gaussian random matrices

Sequential complexities and uniform martingale laws of large numbers

Dimension Reduction by Random Hyperplane Tessellations

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