Metric characterization of random variables and random processes

By V. Buldygin, I. V. Kozachenko. 2000

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Subfields: Management, Monitoring, Policy and Law

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Citations in the corpus

Local Privacy, Data Processing Inequalities, and Statistical Minimax Rates (2013, methodology): “As a consequence, we obtain via standard bounds [11] that”
A ug 2 01 4 Local Privacy , Data Processing Inequalities , and Minimax Rates (2018, methodology): “As a consequence, we obtain via standard bounds [12] that”
Minimax-Optimal Rates For Sparse Additive Models Over Kernel Classes Via Convex Programming (2010, methodology): “ub-Gaussian with parameter √ B +1. This follows since f∗ is bounded by B and using standard results on sums of independent sub-Gaussian random variables (see e.g. Lemma 1.4 in Buldygin and Kozachenko [9]). Therefore P(|y¯n −µ| > t) ≤ 2exp(− nt2 2(B +1) ). Setting A(δn) = {|¯yn −µ| ≤ √ Bδn}, it is”
Joint support recovery under high-dimensional scaling: Benefits and perils of ℓ 1,∞ -regularization (2008, result): “However, we note that qualitatively similar results will hold for any noise distr ibution with sub-Gaussian tails (see the book [1] for more background).”
High-dimensional covariance estimation by minimizing ℓ1-penalized log-determinant divergence (2008, background): “ ν ≤ 8(maxi Σ∗ ii)(1 +4σ2), as required. It only remains to prove Lemma 10. Proof of Lemma 10. If we can obtain a bound B > 0 such that sup m≥2 E(|Zk;ij|m m! 1/m ≤ B, it then follows (Thm. 3.2, [6]) that Zk;ij is sub-exponentialwith parameter2B in the interval (− 1 2B,+ 1 2B). We obtain such a b”
Union support recovery in high-dimensional multivariate regression (2008, background): “ b min := min i2S k ik 2; (13) correspond to the minimal ‘ 2 row-norm of the coecient matrix B over its non-zero rows. We impose the following conditions on the covariance of the design matrix: 2See Buldygin and Kozachenko (2000) for more details on sub-Gaussian random vectors. 6 (A1) Bounded eigens”
On the sub-Gaussianity of the Beta and Dirichlet distributions (2017, background): “Sub-Gaussian inequalities (Buldygin and Kozachenko, 2000; Pisier, 2016) and related concentration inequalities (Boucheron et al.”
Convergence of series of dependent φ-subgaussian random variables (2008, methodology): “Part (b) can be proved in the same way as Buldygin and Kozachenko [ 3, Theorem 5.2 ].”
Sharp Thresholds for High-Dimensional and Noisy Sparsity Recovery Using $\ell _{1}$ -Constrained Quadratic Programming (Lasso) (2009, background): “In addition to the Gaussian case, the class of sub-Gaussian varieties includes any bounded random variable (e.g., Bernoulli, multinomial, uniform), any random variable with strictly log-concave density [2], [24], and any ﬁnite mixture of sub-Gaussian variables.”
High-dimensional analysis of semidefinite relaxations for sparse principal components (2008, unknown): “cused on a Gaussian sampling distribution, parts of our analysis provide suﬃcient conditions for general noise matrices. While qualitatively similar results should hold for sub-Gaussian distributions [5], it would be interesting to characterize how these conditions change as the tail behavior of the”
Restricted isometry properties and nonconvex compressive sensing (2007, methodology): “5 of [26], which for our setting says that”
Randomized Smoothing for Stochastic Optimization (2011, methodology): “For reference purposes, we state here some standard deﬁnitions and facts about sub-Gaussian and sub-exponential random variables (see the books [7, 21, 35] for further details).”