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On the tightness of Gaussian concentration for convex functions

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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.

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References

  1. S. Brazitikos, A. Giannopoulos, P. Valettas and B.-H. Vritsiou, Geometry of Isotropic Convex Bodies, American Mathematical Society, Providence, RI, 2014.

    MATH  Google Scholar 

  2. S. G. Bobkov and C. Houdré, A converse Gaussian Poincaré-type inequality for convex functions, Statist. Probab. Lett. 44 (1999), 281–290.

    MathSciNet  MATH  Google Scholar 

  3. S. Boucheron, G. Lugosi and P. Massart, Concentration Inequalities, Oxford University Press, Oxford, 2013.

    MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. C. Borell, The Ehrhard inequality, C. R. Math. Acad. Sci. Paris 337 (2003), 663–666.

    MathSciNet  MATH  Google Scholar 

  6. 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.

    MATH  Google Scholar 

  7. S. Chatterjee, Superconcentration and Related Topics, Springer, Cham, 2014.

    MATH  Google Scholar 

  8. S. Chatterjee, A general method for lower bounds on fluctuations of random variables, Ann. Probab. 47 (2019), 2140–2171.

    MathSciNet  MATH  Google Scholar 

  9. H. Chernoff. A note on an inequality involving the normal distribution, Ann. Probab. 9 (1981), 533–535.

    MathSciNet  MATH  Google Scholar 

  10. L. H. Y. Chen. An inequality for the multivariate normal distribution, J. Multivariate Anal. 12 (1982), 306–315.

    MathSciNet  MATH  Google Scholar 

  11. 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.

    MATH  Google Scholar 

  12. A. Ehrhard, Symétrisation dans l’espace de Gauss, Math. Scand. 53 (1983), 281–301.

    MathSciNet  MATH  Google Scholar 

  13. A. Ehrhard. Inégalités isopérimétriques et intégrales de Dirichlet gaussiennes, Ann. Sci. École Norm. Sup. (4) 17 (1984), 317–332.

    MathSciNet  MATH  Google Scholar 

  14. A. Ehrhard, Sur l’inégalité de Sobolev logarithmique de Gross, in Seminar on Probability, XVIII, Springer, Berlin, 1984, pp. 194–196.

    MATH  Google Scholar 

  15. Y. Gordon, Some inequalities for Gaussian processes and applications, Israel J. Math. 50 (1985), 265–289.

    MathSciNet  MATH  Google Scholar 

  16. L. Gross, Logarithmic Sobolev inequalities, Amer. J. Math. 97 (1975), 1061–1083.

    MathSciNet  MATH  Google Scholar 

  17. 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.

    MathSciNet  MATH  Google Scholar 

  18. P. Ivanisvili and A. Volberg, Bellman partial differential equation and the hill property for classical isoperimetric problems, arXiv:1506.03409 [math.AP].

  19. 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.

    MATH  Google Scholar 

  20. 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.

    MATH  Google Scholar 

  21. M. Ledoux, The Concentration of Measure Phenomenon, American Mathematical Society, Providence, RI, 2001.

    MATH  Google Scholar 

  22. M. Ledoux and M. Talagrand, Probability in Banach Spaces, Springer-Verlag, Berlin, 1991.

    MATH  Google Scholar 

  23. A. E. Litvak, V. D. Milman and G. Schechtman, Averages of norms and quasi-norms, Math. Ann. 312 (1998), 95–124.

    MathSciNet  MATH  Google Scholar 

  24. 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.

    MathSciNet  Google Scholar 

  25. V. D. Milman and G. Schechtman, Asymptotic Theory of Finite-Dimensional Normed Spaces, Springer-Verlag, Berlin, 1986.

    MATH  Google Scholar 

  26. A. Naor, The surface measure and cone measure on the sphere of l np , Trans. Amer. Math. Soc. 359 (2007), 1045–1079.

    MathSciNet  MATH  Google Scholar 

  27. E. Nelson, Dynamical Theories of Brownian Motion, Princeton University Press, Princeton, NJ, 1967.

    MATH  Google Scholar 

  28. J. Neeman and G. Paouris, An interpolation proof of Ehrhard’s inequality, arXiv:1605.07233 [math.PR].

  29. G. Paouris and P. Valettas, On Dvoretzky’s theorem for subspaces of L p, J. Funct. Anal. 275 (2018), 2225–2252.

    MathSciNet  MATH  Google Scholar 

  30. G. Paouris and P. Valettas, A Gaussian small deviation inequality for convex functions, Ann. Probab. 46 (2018), 1441–1454.

    MathSciNet  MATH  Google Scholar 

  31. G. Paouris and P. Valettas, Dichotomies, structure, and concentration in normed spaces, Adv. Math. 332 (2018), 438–464.

    MathSciNet  MATH  Google Scholar 

  32. G. Paouris and P. Valettas, Variance estimates and almost Euclidean structure, Adv. Geom. 19 (2019), 165–189.

    MathSciNet  MATH  Google Scholar 

  33. G. Paouris, P. Valettas and J. Zinn, Random version of Dvoretzky’s theorem in \(\ell _p^n\), Stochastic Process. Appl. 127 (2017), 3187–3227.

    MathSciNet  MATH  Google Scholar 

  34. M. Rudelson and R. Vershynin, Hanson-Wright inequality and sub-Gaussian concentration, Electron. Commun. Probab. 18 (2013), 2013.

    MathSciNet  MATH  Google Scholar 

  35. 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.

    MATH  Google Scholar 

  36. G. Schechtman, The random version of Dvoretzky’s theorem in ℓ n∞ , in Geometric Aspects of Functional Analysis, Springer, Berlin, 2007, pp. 265–270.

    MATH  Google Scholar 

  37. 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.

    MathSciNet  Google Scholar 

  38. M. Steele, The Paley-Zygnund argument and three variations, Class note; available online, 2004.

  39. Y. Shenfeld and R. van Handel, The equality cases of the Ehrhard-Borell inequality, Adv. Math. 331 (2018), 339–386.

    MathSciNet  MATH  Google Scholar 

  40. 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.

    MATH  Google Scholar 

  41. M. Talagrand, On Russo’s approximate zero-one law, Ann. Probab. 22 (1994), 1576–1587.

    MathSciNet  MATH  Google Scholar 

  42. M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces, Inst. Hautes Études Sci. Publ. Math. 81 (1995), 73–205.

    MathSciNet  MATH  Google Scholar 

  43. K. Tanguy, Some superconcentration inequalities for extrema of stationary Gaussian processes, Statist. Probab. Lett. 106 (2015), 239–246.

    MathSciNet  MATH  Google Scholar 

  44. K. Tanguy, Non-asymptotic variance bounds and deviation inequalities by optimal transport, Electron. J. Probab. 24 (2019), paper no. 13, 18 pp.

  45. K. E. Tikhomirov, The randomized Dvoretzky’s theorem in l n∞ and the χ-distribution, in Geometric Aspects of Functional Analysis, Springer, Cham, 2014, pp. 455–463.

    MATH  Google Scholar 

  46. K. E. Tikhomirov, Superconcentration, and randomized Dvoretzky’s theorem for spaces with 1-unconditional bases, J. Funct. Anal. 274 (2018), 121–151.

    MathSciNet  MATH  Google Scholar 

  47. R. van Handel, The Borell-Ehrhard game, Probab. Theory Related Fields 170 (2018), 555–585.

    MathSciNet  MATH  Google Scholar 

  48. R. van Handel, Private communication, September 2017.

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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.

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Authors and Affiliations

  1. Department of Mathematics, University of Missouri, Columbia, MO, 65211, USA

    Petros Valettas

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  1. Petros Valettas
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Correspondence to Petros Valettas.

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Supported by the NSF grant DMS-1612936.

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Valettas, P. On the tightness of Gaussian concentration for convex functions. JAMA 139, 341–367 (2019). https://doi.org/10.1007/s11854-021-0073-7

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  • DOI: https://doi.org/10.1007/s11854-021-0073-7

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