Abstract
This chapter investigates generic chaining for the supremum of a random process when this process is the convex hull of a simpler one. By geometric arguments a generic chaining result is obtained under eigenvalues conditions. The case where the coefficients of the convex combination have finite entropy is also considered. Moreover, sparse approximations of convex hulls are studied. The problem of deriving the dual norm inequality via generic chaining remains open.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
There is a small clash of notation. In this section S is throughout the index of the last generation, and is not to be confused with an active set S β , \(\beta \in \mathbb{R}^{p}\).
References
P. Bühlmann, S. van de Geer, Statistics for High-Dimensional Data: Methods, Theory and Applications (Springer, Heidelberg, 2011)
B. Carl, Inequalities of Bernstein-Jackson-type and the degree of compactness of operators in banach spaces. Ann. Inst. Fourier 35, 79–118 (1985)
M. Talagrand, The Generic Chaining (Springer, Heidelberg, 2005)
R. van Handel, Chaining, interpolation and convexity. ArXiv:1508.05906 (2015)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2016 Springer International Publishing Switzerland
About this chapter
Cite this chapter
van de Geer, S. (2016). Metric Structure of Convex Hulls. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_18
Download citation
DOI: https://doi.org/10.1007/978-3-319-32774-7_18
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-32773-0
Online ISBN: 978-3-319-32774-7
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)