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Empirical Processes with a Bounded Ψ 1 Diameter

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Abstract

We study the empirical process \({{\rm sup}_{f \in F}|N^{-1}\sum_{i=1}^{N}\,f^{2}(X_i)-\mathbb{E}f^{2}|}\), where F is a class of mean-zero functions on a probability space (Ω, μ), and \({(X_{i})_{i =1}^N}\) are selected independently according to μ.

We present a sharp bound on this supremum that depends on the Ψ 1 diameter of the class F (rather than on the Ψ 2 one) and on the complexity parameter γ 2(F,Ψ 2). In addition, we present optimal bounds on the random diameters \({{\rm sup}_{f \in F} {\rm max}_{|I|=m}\left(\sum_{i \in I} f^{2}(X_{i})\right)^{1/2}}\) using the same parameters. As applications, we extend several well-known results in Asymptotic Geometric Analysis to any isotropic, log-concave ensemble on \({\mathbb{R}^n}\).

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References

  1. Adamczak R., Litvak A., Pajor A., Tomczak-Jaegermann N.: Quantitative estimates of the convergence of the empirical covariance matrix in log-concave ensembles. J. Amer. Math. Soc. 23, 5350–561 (2010)

    Google Scholar 

  2. R. Adamczak, A. Litvak, A. Pajor, N. Tomczak-Jaegermann, Restricted isometry property of matrices with independent columns and neighborly polytopes by random sampling, preprint.

  3. Artstein S.: Change in the diameter of a convex body under a random signprojection, Geometric Aspects of Functional Analysis. Springer Lecture Notes in Math. 1850, 31–39 (2004)

    MathSciNet  Google Scholar 

  4. Aubrun G.: Sampling convex bodies: a random matrix approach. Proc. Amer. Math. Soc. 135, 1293–1303 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  5. Barthe F., Guédon O., Mendelson S., Naor A.: A probabilistic approach to the geometry of the \({\ell_{p}^{n}}\) ball. Ann. Probab. 33(2), 480–513 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  6. P.L. Bartlett, S. Mendelson, J. Neeman, ℓ1-regularized linear regression: Persistence and oracle inequalities, submitted.

  7. Bobkov S.G., Nazarov F.L.: On convex bodies and log-concave probability measures with unconditional basis, Geometric Aspects of Functional Analysis. Springer Lecture Notes in Mathematics 1807, 53–69 (2003)

    MathSciNet  Google Scholar 

  8. Borell C.: The Brunn–Minkowski inequality in Gauss space. Invent. Math. 30, 207–216 (1975)

    Article  MATH  MathSciNet  Google Scholar 

  9. Bourgain J.: Random points in isotropic convex bodies, in “Convex Geometric Analysis (Berkeley, CA, 1996)”. Math. Sci. Res. Inst. Publ. 34, 53–58 (1999)

    MathSciNet  Google Scholar 

  10. Candes E., Romberg J., Tao T.: Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information. IEEE Trans. Info. Theory 52(2), 489–509 (2006)

    Article  MathSciNet  Google Scholar 

  11. Candes E., Tao T.: Near-optimal signal recovery from random projections: universal encoding strategies?. IEEE Trans. Inform. Theory 52(12), 5406–5425 (2006)

    Article  MathSciNet  Google Scholar 

  12. Candes E., Tao T.: The Dantzig selector: statistical estimation when p is much larger than n. Ann. Stat. 35(6), 2313–2351 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  13. R.M. Dudley, Uniform Central Limit Theorems, Cambridge Studies in Advanced Mathematics 63, Cambridge University Press (1999).

  14. Fernique X.: Régularité des trajectoires des fonctiones aléatoires gaussiennes, Ecole d'Eté de Probabilités de St-Flour 1974. Springer Lecture Notes in Mathematics 480, 1–96 (1975)

    Article  MathSciNet  Google Scholar 

  15. A.A. Giannopoulos, Notes on isotropic convex bodies, available at http://users.uoa.gr/~apgiannop/

  16. Giannopoulos A.A., Milman V.D.: Concentration property on probability spaces, Adv. Math. 156, 77–106 (2000)

    MATH  MathSciNet  Google Scholar 

  17. E. Giné, V.H. de la Peña, Decoupling, Springer, 1999.

  18. Giné E., Zinn J.: Some limit theorems for empirical processes, Ann. Probab. 12(4), 929–989 (1984)

    Article  MATH  MathSciNet  Google Scholar 

  19. Greenshtein E., Ritov Y.: Persistence in high-dimensional linear predictor selection and the virtue of overparametrization. Bernoulli 10(6), 971–988 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  20. Guédon O., Rudelson M.: L p moments of random vectors via majorizing measures. Adv. Math. 208(2), 798–823 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Kannan R., Lovász L., Simonovits M.: Random walks and O*(n 5) volume algorithm for convex bodies. Random Structures and Algorithms 2(1), 10–50 (1997)

    Google Scholar 

  22. Klartag B., Mendelson S.: Empirical Processes and Random Projections. Journal of Functional Analysis 225(1), 229–245 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  23. M. Ledoux, M. Talagrand, Probability in Banach Spaces. Isoperimetry and Processes, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) 23, Springer- Verlag, Berlin (1991).

  24. Mendelson S., Pajor A., Tomczak-Jaegermann N.: Reconstruction and subgaussian operators. Geom. Funct. Anal. 17(4), 1248–1282 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  25. Mendelson S.: On weakly bounded empirical processes. Math. Annalen 340(2), 293–314 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  26. V.D. Milman, A note on a low-M* estimate, in “Geometry of Banach spaces (Strobl, 1989)”, London Math. Soc. Lecture Notes Ser. 158, Cambridge University Press (1990), 219–229.

  27. V.D. Milman, G. Schechtman, Asymptotic Theory of Finite Dimensional Normed Spaces, Springer Lecture Notes in Mathematics 1200 (1986).

  28. Pajor A., Tomczak-Jaegermann N.: Subspaces of small codimension of finitedimensional Banach spaces. Proceedings of the AMS. 97(4), 637–642 (1986)

    MATH  MathSciNet  Google Scholar 

  29. A. Pajor, N. Tomczak-Jaegermann, Nombres de Gelfand et sections euclidiennes de grande dimension (French) [Gelfand numbers and high-dimensional Euclidean sections] Séminaire d'Analyse Fonctionelle 1984/1985, Publ. Math. Univ. Paris VII, 26, Univ. Paris VII, Paris, (1986), 37–47.

  30. Paouris G.: Concentration of mass on convex bodies. Geom. Funct. Anal. 16(5), 1021–1049 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  31. G. Pisier, The Volume of Convex Bodies and Banach Space Geometry, Cambridge University Press, 1989.

  32. Rudelson M.: Random vectors in the isotropic position. J. Funct. Anal. 164, 60–72 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  33. Rudelson M.: Extremal distances between sections of convex bodies. Geom. Funct. Anal. 14(5), 1063–1088 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  34. Talagrand M.: Regularity of Gaussian processes. Acta Math. 159, 99–149 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  35. Talagrand M.: The supremum of some canonical processes. American Journal of Mathematics 116, 283–325 (1994)

    Article  MATH  MathSciNet  Google Scholar 

  36. M. Talagrand, The Generic Chaining, Springer, 2005.

  37. A.W. Van der Vaart, J.A. Wellner, Weak Convergence and Empirical Processes, Springer Verlag, 1996.

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Authors and Affiliations

  1. Department of Mathematics, Technion, I.I.T, Haifa, 32000, Israel

    Shahar Mendelson

  2. Centre for Mathematics and its Applications, Institute of Advanced Studies, The AustralianNational University, Canberra, ACT, 0200, Australia

    Shahar Mendelson

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  1. Shahar Mendelson
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Mendelson, S. Empirical Processes with a Bounded Ψ 1 Diameter. Geom. Funct. Anal. 20, 988–1027 (2010). https://doi.org/10.1007/s00039-010-0084-5

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  • DOI: https://doi.org/10.1007/s00039-010-0084-5

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