Advertisement

On the tightness of Gaussian concentration for convex functions

  • Petros ValettasEmail author
Article
  • 1 Downloads

Abstract

The concentration of measure phenomenon in Gauss’ space states that every L-Lipschitz map f on ℝn satisfies
$${\gamma _n}\left({\left\{{x:| {f(x) - {M_f}|\,\geqslant t} } \right\}} \right)\,\leqslant 2{e^{- {{{t^2}} \over {2{L^2}}},}}\quad t > 0,$$
where γn is the standard Gaussian measure on ℝn and Mf is a median of f. In this work, we provide necessary and sufficient conditions for when this inequality can be reversed, up to universal constants, in the case when f is additionally assumed to be convex. In particular, we show that if the variance Var(f) (with respect to γn) satisfies \(\alpha L\leqslant \sqrt {{\rm{Var(}}f{\rm{)}}} \) for some 0 < α ⩽ 1, then
$${\gamma _n}\left({\left\{{x:\left| {f(x) - {M_f}} \right|\geqslant t} \right\}} \right)\,\geqslant \,c{e^{- C{{{t^2}} \over {{L^2}}}}},\quad t > 0,$$
where c, C > 0 are constants depending only on α.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to thank Grigoris Paouris for posing him the question about the tightness of the concentration and for many fruitful discussions. He would also like to thank Peter Pivovarov for useful advice and comments, Ramon van Handel and Emanuel Milman for valuable remarks. Thanks also go to the anonymous referee whose helpful comments improved the exposition of this note.

References

  1. [BGVV14]
    S. Brazitikos, A. Giannopoulos, P. Valettas and B.-H. Vritsiou, Geometry of Isotropic Convex Bodies, American Mathematical Society, Providence, RI, 2014.zbMATHCrossRefGoogle Scholar
  2. [BH99]
    S. G. Bobkov and C. Houdré, A converse Gaussian Poincaré-type inequality for convex functions, Statist. Probab. Lett. 44 (1999), 281–290.MathSciNetzbMATHCrossRefGoogle Scholar
  3. [BLM13]
    S. Boucheron, G. Lugosi and P. Massart, Concentration Inequalities, Oxford University Press, Oxford, 2013.zbMATHCrossRefGoogle Scholar
  4. [Bor75]
    C. Borell, The Brunn-Minkowski inequality in Gauss space, Invent. Math. 30 (1975), 207–216.MathSciNetzbMATHCrossRefGoogle Scholar
  5. [Bor03]
    C. Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (2003), 663–666.MathSciNetzbMATHCrossRefGoogle Scholar
  6. [CEL12]
    D. Cordero-Erausquin and M. Ledoux, Hypercontractive measures, Talagrand’s inequality, and influences, in Geometric Aspects of Functional Analysis, Springer, Heidelberg, 2012, pp. 169–189.zbMATHGoogle Scholar
  7. [Cha14]
    S. Chatterjee, Superconcentration and Related Topics, Springer, Cham, 2014.zbMATHCrossRefGoogle Scholar
  8. [Cha19]
    S. Chatterjee, A general method for lower bounds on fluctuations of random variables, Ann. Probab. 47 (2019), 2140–2171.MathSciNetzbMATHCrossRefGoogle Scholar
  9. [Che81]
    H. Chernoff. A note on an inequality involving the normal distribution, Ann. Probab. 9 (1981), 533–535.MathSciNetzbMATHCrossRefGoogle Scholar
  10. [Che82]
    L. H. Y. Chen. An inequality for the multivariate normal distribution, J. Multivariate Anal. 12 (1982), 306–315.MathSciNetzbMATHCrossRefGoogle Scholar
  11. [Dvo61]
    A. Dvoretzky. Some results on convex bodies and Banach spaces, in Proceedings of the International Symposium on Linear Spaces (Jerusalem, 1960), Jerusalem Academic Press, Jerusalem; Pergamon, Oxford, 1961, pp. 123–160.zbMATHGoogle Scholar
  12. [Ehr83]
    A. Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983), 281–301.MathSciNetzbMATHCrossRefGoogle Scholar
  13. [Ehr84a]
    A. Ehrhard. Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. (4) 17 (1984), 317–332.MathSciNetzbMATHCrossRefGoogle Scholar
  14. [Ehr84b]
    A. Ehrhard, Sur l’inégalité de Sobolev logarithmique de Gross, in Seminar on Probability, XVIII, Springer, Berlin, 1984, pp. 194–196.zbMATHGoogle Scholar
  15. [Gor85]
    Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265–289.MathSciNetzbMATHCrossRefGoogle Scholar
  16. [Gro75]
    L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061–1083.MathSciNetzbMATHCrossRefGoogle Scholar
  17. [HW71]
    D. L. Hanson and F. T. Wright, A bound on tail probabilities for quadratic forms in independent random variables, Ann. Math. Statist. 42 (1971), 1079–1083.MathSciNetzbMATHCrossRefGoogle Scholar
  18. [IV15]
    P. Ivanisvili and A. Volberg, Bellman partial differential equation and the hill property for classical isoperimetric problems, arXiv:1506.03409 [math.AP].Google Scholar
  19. [Kwa94]
    S. Kwapień, A remark on the median and the expectation of convex functions of Gaussian vectors, in Probability in Banach spaces, 9 (Sandjberg, 1993), Birkhäuser Boston, Boston, MA, 1994, pp. 271–272.zbMATHGoogle Scholar
  20. [Lat02]
    R. Latala, On some inequalities for Gaussian measures, in Proceedings of the International Congress of Mathematicians, Vol. II (Beijing, 2002), Higher Ed. Press, Beijing, 2002, pp. 813–822.zbMATHGoogle Scholar
  21. [Led01]
    M. Ledoux, The Concentration of Measure Phenomenon, American Mathematical Society, Providence, RI, 2001.zbMATHGoogle Scholar
  22. [LeT91]
    M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, Berlin, 1991.zbMATHCrossRefGoogle Scholar
  23. [LMS98]
    A. E. Litvak, V. D. Milman and G. Schechtman, Averages of norms and quasi-norms, Math. Ann. 312 (1998), 95–124.MathSciNetzbMATHCrossRefGoogle Scholar
  24. [Mil71]
    V. D. Milman, A new proof of A. Dvoretzky’s theorem on cross-sections of convex bodies, Funkcional. Anal. i Priložen. 5 (1971), 28–37.MathSciNetGoogle Scholar
  25. [MS86]
    V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Springer-Verlag, Berlin, 1986.zbMATHGoogle Scholar
  26. [Nao07]
    A. Naor, The surface measure and cone measure on the sphere of l pn, Trans. Amer. Math. Soc. 359 (2007), 1045–1079.MathSciNetzbMATHCrossRefGoogle Scholar
  27. [Nel67]
    E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ, 1967.zbMATHGoogle Scholar
  28. [NP16]
    J. Neeman and G. Paouris, An interpolation proof of Ehrhard’s inequality, arXiv:1605.07233 [math.PR].Google Scholar
  29. [PV18]
    G. Paouris and P. Valettas, On Dvoretzky’s theorem for subspaces of L p, J. Funct. Anal. 275 (2018), 2225–2252.MathSciNetzbMATHCrossRefGoogle Scholar
  30. [PV18]
    G. Paouris and P. Valettas, A Gaussian small deviation inequality for convex functions, Ann. Probab. 46 (2018), 1441–1454.MathSciNetzbMATHCrossRefGoogle Scholar
  31. [PV18a]
    G. Paouris and P. Valettas, Dichotomies, structure, and concentration in normed spaces, Adv. Math. 332 (2018), 438–464.MathSciNetzbMATHCrossRefGoogle Scholar
  32. [PV19b]
    G. Paouris and P. Valettas, Variance estimates and almost Euclidean structure, Adv. Geom. 19 (2019), 165–189.MathSciNetzbMATHCrossRefGoogle Scholar
  33. [PVZ17]
    G. Paouris, P. Valettas and J. Zinn, Random version of Dvoretzky’s theorem in \(\ell _p^n\), Stochastic Process. Appl. 127 (2017), 3187–3227.MathSciNetzbMATHCrossRefGoogle Scholar
  34. [RV13]
    M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab. 18 (2013), 2013.MathSciNetzbMATHCrossRefGoogle Scholar
  35. [Sch89]
    G. Schechtman, A remark concerning the dependence on ∊ in Dvoretzky’s theorem, in Geometric Aspects of Functional Analysis (1987–88), Springer, Berlin, 1989, pp. 274–277.zbMATHGoogle Scholar
  36. [Sch07]
    G. Schechtman, The random version of Dvoretzky’s theorem in ℓ n, in Geometric Aspects of Functional Analysis, Springer, Berlin, 2007, pp. 265–270.zbMATHGoogle Scholar
  37. [ST74]
    V. N. Sudakov and B. S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures, Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI) 41 (1974), 14–24, 165.MathSciNetGoogle Scholar
  38. [Ste04]
    M. Steele, The Paley-Zygnund argument and three variations, Class note; available online, 2004.Google Scholar
  39. [SvH18]
    Y. Shenfeld and R. van Handel, The equality cases of the Ehrhard-Borell inequality, Adv. Math. 331 (2018), 339–386.MathSciNetzbMATHCrossRefGoogle Scholar
  40. [Tal91]
    M. Talagrand, A new isoperimetric inequality and the concentration of measure phenomenon, in Geometric Aspects of Functional Analysis (1989–90), Springer, Berlin, 1991, pp. 94–124.zbMATHGoogle Scholar
  41. [Tal94]
    M. Talagrand, On Russo’s approximate zero-one law, Ann. Probab. 22 (1994), 1576–1587.MathSciNetzbMATHCrossRefGoogle Scholar
  42. [Tal95]
    M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 73–205.MathSciNetzbMATHCrossRefGoogle Scholar
  43. [Tan15]
    K. Tanguy, Some superconcentration inequalities for extrema of stationary Gaussian processes, Statist. Probab. Lett. 106 (2015), 239–246.MathSciNetzbMATHCrossRefGoogle Scholar
  44. [Tan19]
    K. Tanguy, Non-asymptotic variance bounds and deviation inequalities by optimal transport, Electron. J. Probab. 24 (2019), paper no. 13, 18 pp.Google Scholar
  45. [Tik14]
    K. E. Tikhomirov, The randomized Dvoretzky’s theorem in l nand the χ-distribution, in Geometric Aspects of Functional Analysis, Springer, Cham, 2014, pp. 455–463.zbMATHGoogle Scholar
  46. [Tik18]
    K. E. Tikhomirov, Superconcentration, and randomized Dvoretzky’s theorem for spaces with 1-unconditional bases, J. Funct. Anal. 274 (2018), 121–151.MathSciNetzbMATHCrossRefGoogle Scholar
  47. [vH18a]
    R. van Handel, The Borell-Ehrhard game, Probab. Theory Related Fields 170 (2018), 555–585.MathSciNetzbMATHCrossRefGoogle Scholar
  48. [vH17b]
    R. van Handel, Private communication, September 2017.Google Scholar

Copyright information

© The Hebrew University of Jerusalem 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of MissouriColumbiaUSA

Personalised recommendations