Abstract
This chapter presents probability inequalities for the (dual) norm of a Gaussian vector in \(\mathbb{R}^{p}\). For Gaussian vectors there are ready-to-use concentration Borell, C. inequalities (e.g. Borell, 1975). Here however, results are derived using direct arguments. The extension to for example sub-Gaussian vectors is then easier to read off. Bounds for the supremum are given, for the ℓ 2-norm, and more generally for dual norms of norms generated from a convex cone.
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References
L. Birgé, P. Massart, Minimum contrast estimators on sieves: exponential bounds and rates of convergence. Bernoulli 4, 329–375 (1998)
C. Borell, The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)
P. Bühlmann, S. van de Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications (Springer, Heidelberg, 2011)
B. Laurent, P. Massart, Adaptive estimation of a quadratic functional by model selection. Ann. Stat. 28, 1302–1338 (2000)
K. Lounici, M. Pontil, S. van de Geer, A. Tsybakov, Oracle inequalities and optimal inference under group sparsity. Ann. Stat. 39, 2164–2204 (2011)
A. Maurer, M. Pontil, Structured sparsity and generalization. J. Mach. Learn. Res. 13, 671–690 (2012)
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van de Geer, S. (2016). Empirical Process Theory for Dual Norms. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_8
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DOI: https://doi.org/10.1007/978-3-319-32774-7_8
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