High dimensional probability : proceedings of the Fourth International Conference

By Evarist Gin'e, Vladimir Koltchinskii, Wenbo V. Li, Joel C. Zinn. 2006

377 citations recorded by the source corpus.

About forty years ago it was realized by several researchers that the essential features of certain objects of Probability theory, notably Gaussian processes and limit theorems, may be better understood if they are considered in settings that do not impose structures extraneous to the problems at hand. For instance, in the case of sample continuity and boundedness of Gaussian processes, the essential feature is the metric or pseudometric structure induced on the index set by the covariance structure of the process, regardless of what the index set may be. This point of view ultimately led to the Fernique-Talagrand majorizing measure characterization of sample boundedness and continuity of Gaussian processes, thus solving an important problem posed by Kolmogorov. Similarly, separable Banach spaces provided a minimal setting for the law of large numbers, the central limit theorem and the law of the iterated logarithm, and this led to the elucidation of the minimal (necessary and/or sufficient) geometric properties of the space under which different forms of these theorems hold. However, in light of renewed interest in Empirical processes, a subject that has considerably influenced modern Statistics, one had to deal with a non-separable Banach space, namely $\mathcal{L}_{\infty}$. With separability discarded, the techniques developed for Gaussian processes and for limit theorems and inequalities in separable Banach spaces, together with combinatorial techniques, led to powerful inequalities and limit theorems for sums of independent bounded processes over general index sets, or, in other words, for general empirical processes.

Subfields: Statistics and Probability

View Paper: https://www.semanticscholar.org/paper/629ac1415536338db70d08ca8569ea7e0246ff88

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Citations in the corpus

Provable Filter Pruning for Efficient Neural Networks (2019, methodology): “Theorem 1 (Bernstein’s inequality (Vershynin, 2016 Then, for every ε ≥ 0 , δ ∈ (0 , 1) , we have that ] be arbitrary and applying Theorem 1 to the mean of the random variables , we observe that the number of samples required for a sufﬁciently high concentration around the mean is highly dependent…”
Dynamics of stochastic gradient descent for two-layer neural networks in the teacher–student setup (2019, neutral): “(cid:80) We ﬁrst note that the sum of the output noise variables ζ µ is a simple sum over uncorrelated, (sub-) Gaussian random variables rescaled by 1 /N and thus by Hoeffding’s inequality almost surely smaller than a constant 49 .”
Are adversarial examples inevitable? (2018, background): “The classical isoperimetric inequality is a simple geometric statement, and frequently appears without absolute bounds on the size of the (cid:15) -expansion of a half-sphere, or with bounds that involve unspeciﬁed constants (Vershynin, 2017).”
A Donsker Theorem for Lévy Measures (2012, background): “This hope is nourished by the phenomenon – first observed, in a general empirical process setting unrelated to the present situation, by Radulović and Wegkamp (2000), and recently developed further in several directions by Giné and Nickl (2008) – that smoothed empirical processes may converge in sit”
Chaining Mutual Information and Tightening Generalization Bounds (2018, methodology): “More specifically, chaining is the method for proving the tightest generalization bound using VC-dimension [7], [8].”
High‐dimensional quantile regression: Convolution smoothing and concave regularization (2021, background): “11 in Vershynin (2018)—Sudakov-Fernique’s Gaussian comparison inequality, we obtain”
Distance geometry and data science (2019, neutral): “2015; Vershynin 2018).”
Learning with Bad Training Data via Iterative Trimmed Loss Minimization (2018, background): “Notice that we can choose an (cid:15) -net in R d , which includes (1 + 2 ) d points [Ver16].”
Data-Dependent Coresets for Compressing Neural Networks with Applications to Generalization Bounds (2018, methodology): “variables, which implies that E [Y] = 0 and that we can readily apply Bernstein’s inequality for subexponential random variables [46] to obtain for t ≥ 0”
Concentration inequalities for random tensors (2019, positive): “Throughout this paper, we use basic facts about subgaussian and subexponential random variables that can be found e.g. in [35, Chapters 2– 3 ] 2.1.”
Spectral Deconfounding and Perturbed Sparse Linear Models (2018, background): “In both cases we have λ min (Γ) ≥ λ min (Ω 1 / 2 ) λ min ( Z ) = Ω p ( √ p ) since λ min (Ω) is bounded from below and λ min ( Z ) = Ω p ( √ p ) (see Corollary 5.35 from [25]). δ = O p ( √ q ) follows from central limit theorem when the components of δ are i.i.d.”