Abstract
This chapter reviews symmetrization results, presents the contraction theorem of Ledoux and Talagrand (1991) and a multivariate extension and also the concentration (or actually the “deviation” part of) theorems of Bousquet (2002) and Massart (2000). The chapter gathers a collection of tools which have been used in Chap. 10 No proofs are given except for Theorem 16.3 on the multivariate contraction inequality (as it is not a standard formulation) and Lemma 16.2 (as this proof connects well with results from Chap. 8).
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Notes
- 1.
All statements are true “modulo measurability”: the quantities involved may not be measurable. Easiest way out is to replace the supremum over an uncountable class by one over a countable class.
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van de Geer, S. (2016). Symmetrization, Contraction and Concentration. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_16
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DOI: https://doi.org/10.1007/978-3-319-32774-7_16
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