Advertisement

Concentration Properties of Restricted Measures with Applications to Non-Lipschitz Functions

  • Sergey G. Bobkov
  • Piotr Nayar
  • Prasad Tetali
Chapter
Part of the Lecture Notes in Mathematics book series (LNM, volume 2169)

Abstract

We show that, for any metric probability space (M, d, μ) with a subgaussian constant σ2(μ) and any Borel measurable set A ⊂ M, we have \(\sigma ^{2}(\mu _{A}) \leq c\log \left (e/\mu (A)\right )\sigma ^{2}(\mu )\), where μ A is a normalized restriction of μ to the set A and c is a universal constant. As a consequence, we deduce concentration inequalities for non-Lipschitz functions.

2010 Mathematics Subject Classification.

Primary 60Gxx 

Notes

Acknowledgements

The authors gratefully acknowledge the support and hospitality of the Institute for Mathematics and its Applications, and the University of Minnesota, Minneapolis, where much of this work was conducted. The second named author would like to acknowledge the hospitality of the Georgia Institute of Technology, Atlanta, during the period 02/8-13/2015.

We would like to thank anonymous referee for pointing out to us several relevant references. S.G. Bobkov’s research supported in part by the Humboldt Foundation, NSF and BSF grants. P. Nayar’s research supported in part by NCN grant DEC-2012/05/B/ST1/00412. P. Tetali’s research supported in part by NSF DMS-1407657.

References

  1. 1.
    S. Aida, D. Strook, Moment estimates derived from Poincaré and logarithmic Sobolev inequalities. Math. Res. Lett. 1, 75–86 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    N. Alon, R. Boppana, J. Spencer, An asymptotic isoperimetric inequality. Geom. Funct. Anal. 8, 411–436 (1998)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    D. Bakry, M. Ledoux, Lévy–Gromov’s isoperimetric inequality for an infinite dimensional diffusion generator. Invent. Math. 123, 259–281 (1996)MathSciNetzbMATHGoogle Scholar
  4. 4.
    F. Barthe, E. Milman, Transference principles for log-Sobolev and spectral-gap with applications to conservative spin systems. Commun. Math. Phys. 323 (2), 575–625 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    S.G. Bobkov, Localization proof of the isoperimetric Bakry-Ledoux inequality and some applications. Teor. Veroyatnost. i Primenen. 47 (2), 340–346 (2002) Translation in: Theory Probab. Appl. 47 (2), 308–314 (2003)Google Scholar
  6. 6.
    S.G. Bobkov, Perturbations in the Gaussian isoperimetric inequality. J. Math. Sci. 166 (3), 225–238 (2010). New York. Translated from: Problems in Mathematical Analysis, vol. 45 (2010), pp. 3–14Google Scholar
  7. 7.
    S.G. Bobkov, F. Götze, Exponential integrability and transportation cost related to logarithmic Sobolev inequalities. J. Funct. Anal. 163 (1), 1–28 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    S.G. Bobkov, C. Houdré, P. Tetali, The subgaussian constant and concentration inequalities. Isr. J. Math. 156, 255–283 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    J. Ding, E. Mossel, Mixing under monotone censoring. Electron. Commun. Probab. 19, 1–6 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    M.D. Kirszbraun, Über die zusammenziehende und Lipschitzsche Transformationen. Fund. Math. 22, (1934) 77–108.zbMATHGoogle Scholar
  11. 11.
    M. Ledoux, The Concentration of Measure Phenomenon. Mathematical Surveys and Monographs, vol. 89 (American Mathematical Society, Providence, RI, 2001)Google Scholar
  12. 12.
    K. Marton, Bounding \(\bar{d}\)-distance by informational divergence: a method to prove measure concentration. Ann. Probab. 24 (2), 857–866 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    B. Maurey, Some deviation inequalities. Geom. Funct. Anal. 1, 188–197 (1991)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    E.J. McShane, Extension of range of functions. Bull. Am. Math. Soc. 40 (12), 837–842 (1934)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    E. Milman, On the role of convexity in isoperimetry, spectral gap and concentration. Invent. Math. 177 (1), 1–43 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    E. Milman, Properties of isoperimetric, functional and transport-entropy inequalities via concentration. Probab. Theory Relat. Fields 152 (3–4), 475–507 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    C.E. Mueller, F.B. Weissler, Hypercontractivity for the heat semigroup for ultraspherical polynomials and on the n-sphere. J. Funct. Anal. 48, 252–283 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M. Talagrand, An isoperimetric theorem on the cube and the Khinchine–Kahane inequalities. Proc. Am. Math. Soc. 104, 905–909 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    M. Talagrand, Concentration of measure and isoperimetric inequalities in product spaces. Publ. Math. I.H.E.S. 81, 73–205 (1995)Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.School of MathematicsUniversity of MinnesotaMinneapolisUSA
  2. 2.Institute of Mathematics and its ApplicationsMinneapolisUSA
  3. 3.School of Mathematics and School of Computer ScienceGeorgia Institute of TechnologyAtlantaUSA

Personalised recommendations