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Rearrangement and Prékopa–Leindler Type Inequalities

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High Dimensional Probability VIII

Part of the book series: Progress in Probability ((PRPR,volume 74))

Abstract

We investigate the interactions of functional rearrangements with Prékopa–Leindler type inequalities. It is shown that certain set theoretic rearrangement inequalities can be lifted to functional analogs, thus demonstrating that several important integral inequalities tighten on functional rearrangement about “isoperimetric” sets with respect to a relevant measure. Applications to the Borell–Brascamp–Lieb, Borell–Ehrhard, and the recent polar Prékopa–Leindler inequalities are demonstrated. It is also proven that an integrated form of the Gaussian log-Sobolev inequality sharpens on rearrangement.

A portion of this work relevant to information theory was announced at 56th Annual Allerton Conference on Communication, Control, and Computing [43].

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Notes

  1. 1.

    This follows from the fact that Borel sets are analytic, see [28], and analytic sets are closed under summation and universally measurable.

  2. 2.

    Note that when \(f = \frac {d\nu }{d\mu }\) is the density function of a probability measure ν with respect to μ, H μ(f) is the Kullback–Liebler divergence D(ν||μ) or relative entropy [22].

References

  1. S. Artstein-Avidan, D. Florentin, A. Segal, Polar Prékopa–Leindler inequalities (2017). arXiv preprint. arXiv:1707.08732

    Google Scholar 

  2. F. Barthe, On a reverse form of the Brascamp-Lieb inequality. Invent. Math. 134(2), 335–361 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  3. F. Barthe, N. Huet, On Gaussian Brunn–Minkowski inequalities. Stud. Math. 191(3), 283–304 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  4. W. Beckner, Inequalities in Fourier analysis. Ann. Math. 102(1), 159–182 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  5. N.M. Blachman, The convolution inequality for entropy powers. IEEE Trans. Inf. Theory 11, 267–271 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  6. S.G. Bobkov, G.P. Chistyakov, Bounds for the maximum of the density of the sum of independent random variables. Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. 408, 62–73, 324 (2012)

    Google Scholar 

  7. S.G. Bobkov, G.P. Chistyakov, Entropy power inequality for the Rényi entropy. IEEE Trans. Inf. Theory 61(2), 708–714 (2015)

    Article  MATH  Google Scholar 

  8. S.G. Bobkov, M. Ledoux, From Brunn-Minkowski to Brascamp-Lieb and to logarithmic Sobolev inequalities. Geom. Funct. Anal. 10(5), 1028–1052 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. S.G. Bobkov, M. Ledoux, Weighted Poincaré-type inequalities for Cauchy and other convex measures. Ann. Probab. 37(2), 403–427 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  10. S. Bobkov, A. Marsiglietti, Variants of the entropy power inequality. IEEE Trans. Inf. Theory 63(12), 7747–7752 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Bobkov, J. Melbourne, Hyperbolic measures on infinite dimensional spaces. Probab. Surv. 13, 57–88 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  12. S.G. Bobkov, I. Gentil, M. Ledoux, Hypercontractivity of Hamilton–Jacobi equations. J. Math. Pures Appl. 80(7), 669–696 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. C. Borell, Convex measures on locally convex spaces. Ark. Mat. 12, 239–252 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  14. C. Borell, The Brunn-Minkowski inequality in Gauss space. Invent. Math. 30(2), 207–216 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  15. C. Borell, Convex set functions in d-space. Period. Math. Hungar. 6(2), 111–136 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  16. C. Borell, Inequalities of the Brunn-Minkowski type for Gaussian measures. Probab. Theory Relat. Fields 140(1–2), 195–205 (2008)

    MathSciNet  MATH  Google Scholar 

  17. H.J. Brascamp, E.H. Lieb, Best constants in Young’s inequality, its converse, and its generalization to more than three functions. Adv. Math. 20(2), 151–173 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  18. H.J. Brascamp, E.H. Lieb, On extensions of the Brunn-Minkowski and Prékopa-Leindler theorems, including inequalities for log concave functions, and with an application to the diffusion equation. J. Funct. Anal. 22(4), 366–389 (1976)

    Article  MATH  Google Scholar 

  19. H.J. Brascamp, E.H. Lieb, J.M. Luttinger, A general rearrangement inequality for multiple integrals. J. Funct. Anal. 17, 227–237 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. E.A. Carlen, Superadditivity of Fisher’s information and logarithmic Sobolev inequalities. J. Funct. Anal. 101(1), 194–211 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  21. M.H.M. Costa, T.M. Cover, On the similarity of the entropy power inequality and the Brunn-Minkowski inequality. IEEE Trans. Inf. Theory 30(6), 837–839 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  22. T.M. Cover, J.A. Thomas, Elements of Information Theory (Wiley, New York, 1991)

    Book  MATH  Google Scholar 

  23. A. Dembo, T.M. Cover, J.A. Thomas, Information-theoretic inequalities. IEEE Trans. Inf. Theory 37(6), 1501–1518 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  24. A. Ehrhard, Symétrisation dans l’espace de Gauss. Math. Scand. 53(2), 281–301 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Gozlan, C. Roberto, P.M. Samson, P. Tetali, Displacement convexity of entropy and related inequalities on graphs. Probab. Theory Relat. Fields 160(1–2), 47–94 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  26. N. Gozlan, C. Roberto, P.M. Samson, P. Tetali, Transport proofs of some discrete variants of the Prékopa-Leindler inequality (2019). arXiv preprint. arXiv:1905.04038

    Google Scholar 

  27. L. Gross, Logarithmic Sobolev inequalities. Am. J. Math. 97(4), 1061–1083 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Kechris, Classical Descriptive Set Theory, vol. 156 (Springer Science & Business Media, New York, 2012)

    MATH  Google Scholar 

  29. J. Li, Rényi entropy power inequality and a reverse (2017). Preprint. arXiv:1704.02634

    Google Scholar 

  30. J. Li, J. Melbourne, Further investigations of the maximum entropy of the sum of two dependent random variables, in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2018), pp. 1969–1972

    Google Scholar 

  31. J. Li, A. Marsiglietti, J. Melbourne, Entropic central limit theorem for rényi entropy, in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2019)

    Google Scholar 

  32. J. Li, A. Marsiglietti, J. Melbourne, Further investigations of Rényi entropy power inequalities and an entropic characterization of s-concave densities, in Geometric Aspects of Functional Analysis – Israel Seminar (GAFA) 2017–2019. Lecture Notes in Mathematics, vol. 2256 (to appear)

    Google Scholar 

  33. J. Li, A. Marsiglietti, J. Melbourne, Rényi entropy power inequalities for s-concave densities, in 2019 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2019)

    Google Scholar 

  34. E.H. Lieb, Proof of an entropy conjecture of Wehrl. Commun. Math. Phys. 62(1), 35–41 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  35. M. Madiman, J. Melbourne, P. Xu, Forward and reverse entropy power inequalities in convex geometry, in Convexity Concentration (Springer, New York, 2017), pp. 427–485

    MATH  Google Scholar 

  36. M. Madiman, J. Melbourne, P. Xu, Rogozin’s convolution inequality for locally compact groups (2017). Preprint, arXiv:1705.00642

    Google Scholar 

  37. M. Madiman, L. Wang, J.O. Woo, Entropy inequalities for sums in prime cyclic groups (2017). arXiv preprint. arXiv:1710.00812

    Google Scholar 

  38. M. Madiman, L. Wang, J.O. Woo, Majorization and Rényi entropy inequalities via Sperner theory. Discrete Math. 342, 2911–2923 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  39. A. Marsiglietti, J. Melbourne, A Rényi entropy power inequality for log-concave vectors and parameters in [0, 1], in 2018 IEEE International Symposium on Information Theory (ISIT) (IEEE, Piscataway, 2018), pp. 1964–1968

    Google Scholar 

  40. A. Marsiglietti, J. Melbourne, On the entropy power inequality for the Rényi entropy of order [0, 1]. IEEE Trans. Inf. Theory 65, 1387–1396 (2019)

    Article  MATH  Google Scholar 

  41. J. Martín, M. Milman, Isoperimetry and symmetrization for logarithmic Sobolev inequalities. J. Funct. Anal. 256(1), 149–178 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  42. J. Martín, M. Milman, Pointwise symmetrization inequalities for Sobolev functions and applications. Adv. Math. 225(1), 121–199 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  43. J. Melbourne, Rearrangements and information theoretic inequalities, in 2018 56th Annual Allerton Conference on Communication, Control, and Computing (Allerton) (IEEE, Piscataway, 2018)

    Google Scholar 

  44. Y. Ollivier, C. Villani, A curved Brunn–Minkowski inequality on the discrete hypercube, or: what is the Ricci curvature of the discrete hypercube? SIAM J. Discrete Math. 26(3), 983–996 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  45. E. Ram, I. Sason, On rényi entropy power inequalities. IEEE Trans. Inf. Theory 62(12), 6800–6815 (2016)

    Article  MATH  Google Scholar 

  46. A. Rényi, On measures of entropy and information, in Proceedings of the Fourth Berkeley Symposium on Mathematical Statistics and Probability, Volume 1 (University of California Press, Berkeley, 1961), pp. 547–561

    Google Scholar 

  47. C.E. Shannon, A mathematical theory of communication. Bell Syst. Tech. J. 27, 379–423, 623–656 (1948)

    Article  MathSciNet  MATH  Google Scholar 

  48. A.J. Stam, Some inequalities satisfied by the quantities of information of Fisher and Shannon. Inf. Control 2, 101–112 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  49. V.N. Sudakov, B.S. Tsirel’son, Extremal properties of half-spaces for spherically invariant measures. Zap. Nauch. Sem. LOMI 41, 14–24 (1974). Translated in J. Soviet Math. 9, 9–18 (1978)

    Google Scholar 

  50. L. Wang, M. Madiman, Beyond the entropy power inequality, via rearrangements. IEEE Trans. Inf. Theory 60(9), 5116–5137 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  51. R. Zamir, M. Feder, A generalization of the entropy power inequality with applications. IEEE Trans. Inf. Theory 39(5), 1723–1728 (1993)

    Article  MATH  Google Scholar 

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Authors and Affiliations

  1. Electrical and Computer Engineering, University of Minnesota, Minneapolis, MN, USA

    James Melbourne

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  1. James Melbourne
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Correspondence to James Melbourne .

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Editors and Affiliations

  1. MAP 5, Université Paris Descartes, Paris, France

    Nathael Gozlan

  2. Institute of Mathematics, University of Warsaw, Warsaw, Poland

    Rafał Latała

  3. Centre de Mathématiques Appliquées, Ecole Polytechnique, Palaiseau, France

    Karim Lounici

  4. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Mokshay Madiman

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Melbourne, J. (2019). Rearrangement and Prékopa–Leindler Type Inequalities. In: Gozlan, N., Latała, R., Lounici, K., Madiman, M. (eds) High Dimensional Probability VIII. Progress in Probability, vol 74. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-26391-1_7

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