On Multiplier Processes Under Weak Moment Assumptions

Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)

Abstract

We show that if \(V \subset \mathbb{R}^{n}\) satisfies a certain symmetry condition that is closely related to unconditionality, and if X is an isotropic random vector for which \(\|\big< X,t\big >\| _{L_{p}} \leq L\sqrt{p}\) for every t ∈ Sn−1 and every \(1 \leq p\lesssim \log n\), then the suprema of the corresponding empirical and multiplier processes indexed by V behave as if X were L-subgaussian.

Notes

Acknowledgements

S. Mendelson is supported in part by the Israel Science Foundation.

References

  1. 1.
    F. Albiac, N.J. Kalton, Topics in Banach Space Theory. Graduate Texts in Mathematics, vol. 233 (Springer, New York, 2006)Google Scholar
  2. 2.
    S. Artstein-Avidan, A. Giannopoulos, V.D. Milman, Asymptotic Geometric Analysis. Part I. Mathematical Surveys and Monographs, vol. 202 (American Mathematical Society, Providence, RI, 2015)Google Scholar
  3. 3.
    P. Bühlmann, S. van de Geer, Statistics for High-Dimensional Data. Methods, Theory and Applications. Springer Series in Statistics (Springer, Heidelberg, 2011)Google Scholar
  4. 4.
    S. Foucart, H. Rauhut, A Mathematical Introduction to Compressive Sensing. Applied and Numerical Harmonic Analysis (Birkhäuser/Springer, New York, 2013)CrossRefMATHGoogle Scholar
  5. 5.
    V. Koltchinskii, Oracle Inequalities in Empirical Risk Minimization and Sparse Recovery Problems. Lecture Notes in Mathematics, vol. 2033 (Springer, Heidelberg, 2011). Lectures from the 38th Probability Summer School held in Saint-Flour, 2008, École d’Été de Probabilités de Saint-Flour. [Saint-Flour Probability Summer School]Google Scholar
  6. 6.
    G. Lecué, S. Mendelson, Sparse recovery under weak moment assumptions. Technical report, CNRS, Ecole Polytechnique and Technion (2014). J. Eur. Math. Soc. 19 (3), 881–904 (2017)Google Scholar
  7. 7.
    G. Lecué, S. Mendelson, Regularization and the small-ball method I: sparse recovery. Technical report, CNRS, ENSAE and Technion, I.I.T. (2015). Ann. Stati. (to appear)Google Scholar
  8. 8.
    M. Ledoux, M. Talagrand, Probability in Banach Spaces. Isoperimetry and processes. Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 23 (Springer, Berlin, 1991)Google Scholar
  9. 9.
    S. Mendelson, Learning without concentration for general loss function. Technical report, Technion, I.I.T. (2013). arXiv:1410.3192Google Scholar
  10. 10.
    S. Mendelson, A remark on the diameter of random sections of convex bodies, in Geometric Aspects of Functional Analysis, Lecture Notes in Mathematics, vol. 2116, pp. 395–404 (Springer, Cham, 2014)Google Scholar
  11. 11.
    S. Mendelson, Upper bounds on product and multiplier empirical processes. Stoch. Process. Appl. 126 (12), 3652–3680 (2016)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    S. Mendelson, A. Pajor, N. Tomczak-Jaegermann, Reconstruction and subgaussian operators in asymptotic geometric analysis. Geom. Funct. Anal. 17 (4), 1248–1282 (2007)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    S. Mendelson, G. Paouris, On generic chaining and the smallest singular value of random matrices with heavy tails. J. Funct. Anal. 262 (9), 3775–3811 (2012)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    V.D. Milman, Random subspaces of proportional dimension of finite-dimensional normed spaces: approach through the isoperimetric inequality, in Banach Spaces (Columbia, MO, 1984). Lecture Notes in Mathematics, vol. 1166, pp. 106–115 (Springer, Berlin, 1985)Google Scholar
  15. 15.
    A. Pajor, N. Tomczak-Jaegermann, Nombres de Gel′ fand et sections euclidiennes de grande dimension, in Séminaire d’Analyse Fonctionelle 1984/1985. Publ. Math. Univ. Paris VII, vol. 26, pp. 37–47 (University of Paris VII, Paris, 1986)Google Scholar
  16. 16.
    A. Pajor, N. Tomczak-Jaegermann, Subspaces of small codimension of finite-dimensional Banach spaces. Proc. Am. Math. Soc. 97 (4), 637–642 (1986)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    G. Pisier, The Volume of Convex Bodies and Banach Space Geometry. Cambridge Tracts in Mathematics, vol. 94 (Cambridge University Press, Cambridge, 1989)Google Scholar
  18. 18.
    M. Talagrand, Regularity of Gaussian processes. Acta Math. 159 (1–2), 99–149 (1987)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    M. Talagrand, Upper and Lower Bounds for Stochastic Processes. Modern Methods and Classical Problems. Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 60. (Springer, Heidelberg, 2014)Google Scholar
  20. 20.
    A.W. van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes. With Applications to Statistics. Springer Series in Statistics (Springer, New York, 1996)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.Department of MathematicsTechnion - I.I.T.HaifaIsrael
  2. 2.Mathematical Sciences InstituteThe Australian National UniversityCanberraAustralia

Personalised recommendations