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Lower Bounds for Sparse Quadratic Forms

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

Abstract

Lower bounds for sparse quadratic forms are studied. This has its implications for effective sparsity (or compatibility constants): the effective sparsity with empirical semi-norm \(\|X \cdot \|_{n}\) is bounded in terms of the effective sparsity with theoretical semi-norm \(\|X\cdot \|\). The results are an extension of van de Geer and Muro (Electron. J. Stat. 8:3031–3061, 2014) to more general sparsity inducing norms Ω.

Keywords

  • Lower Bound
  • Effective Sparsity
  • Compatibility Constant
  • Rademacher Sequence
  • Uniform Constant

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References

  • O. Bousquet, A Bennet concentration inequality and its application to suprema of empirical processes. C. R. Acad. Sci. Paris 334, 495–550 (2002)

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  • G. Lecué, S. Mendelson, Sparse recovery under weak moment assumptions. J. Eur. Math. Soc. To appear, available at arXiv preprint. arXiv:1401.2188 (2014)

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  • S. van de Geer, A. Muro, On higher order isotropy conditions and lower bounds for sparse quadratic forms. Electron. J. Stat. 8, 3031–3061 (2014)

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© 2016 Springer International Publishing Switzerland

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van de Geer, S. (2016). Lower Bounds for Sparse Quadratic Forms. In: Estimation and Testing Under Sparsity. Lecture Notes in Mathematics(), vol 2159. Springer, Cham. https://doi.org/10.1007/978-3-319-32774-7_15

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