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Metric Structure of Convex Hulls

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Part of the Lecture Notes in Mathematics book series (LNMECOLE,volume 2159)

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.

Keywords

  • Convex Hull
  • Generic Chain
  • Unbiased Estimator
  • Universal Constant
  • Multinomial Distribution

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Notes

  1. 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

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  • M. Talagrand, The Generic Chaining (Springer, Heidelberg, 2005)

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  • R. van Handel, Chaining, interpolation and convexity. ArXiv:1508.05906 (2015)

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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

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