Skip to main content
SpringerLink
Log in

Optimal Concentration of Information Content for Log-Concave Densities

  • Conference paper
  • First Online:
High Dimensional Probability VII

Part of the book series: Progress in Probability ((PRPR,volume 71))

Abstract

An elementary proof is provided of sharp bounds for the varentropy of random vectors with log-concave densities, as well as for deviations of the information content from its mean. These bounds significantly improve on the bounds obtained by Bobkov and Madiman (Ann Probab 39(4):1528–1543, 2011).

Mathematics Subject Classification (2010). Primary 52A40; Secondary 60E15, 94A17

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. G. Bennett, Probability inequalities for the sum of independent random variables. J. Am. Stat. Assoc. 57 (297), 33–45 (1962)

    Article  MATH  Google Scholar 

  2. S. Bobkov, M. Madiman, Concentration of the information in data with log-concave distributions. Ann. Probab. 39 (4), 1528–1543 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  3. S. Bobkov, M. Madiman, Dimensional behaviour of entropy and information. C. R. Acad. Sci. Paris Sér. I Math. 349, 201–204 Février (2011)

    Google Scholar 

  4. S. Bobkov, M. Madiman, The entropy per coordinate of a random vector is highly constrained under convexity conditions. IEEE Trans. Inform. Theory 57 (8), 4940–4954 (2011)

    Article  MathSciNet  Google Scholar 

  5. S. Bobkov, M. Madiman, Reverse Brunn-Minkowski and reverse entropy power inequalities for convex measures. J. Funct. Anal. 262, 3309–3339 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  6. S. Bobkov, M. Fradelizi, J. Li, M. Madiman, When can one invert Hölder’s inequality? (and why one may want to). Preprint (2016)

    Google Scholar 

  7. F. Bolley, I. Gentil, A. Guillin, Dimensional improvements of the logarithmic Sobolev, Talagrand and Brascamp-Lieb inequalities. Preprint, arXiv:1507:01086 (2015)

    Google Scholar 

  8. C. Borell, Complements of Lyapunov’s inequality. Math. Ann. 205, 323–331 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  9. S. Boucheron, G. Lugosi, P. Massart, Concentration Inequalities (Oxford University Press, Oxford, 2013). A nonasymptotic theory of independence, With a foreword by Michel Ledoux

    Google Scholar 

  10. S. Boyd, L. Vandenberghe, Convex Optimization (Cambridge University Press, Cambridge, 2004)

    Book  MATH  Google Scholar 

  11. D. Cordero-Erausquin, M. Fradelizi, G. Paouris, P. Pivovarov, Volume of the polar of random sets and shadow systems. Math. Ann. 362, 1305–1325 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. R. Eldan, Thin shell implies spectral gap up to polylog via a stochastic localization scheme. Geom. Funct. Anal. 23 (2), 532–569 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  13. R. Eldan, B. Klartag, Approximately Gaussian marginals and the hyperplane conjecture, in Concentration, Functional Inequalities and Isoperimetry, ed. by C. Houdré, M. Ledoux, E. Milman, M. Milman. Contemporary Mathematics, vol. 545 (American Mathematical Society, Providence, RI, 2011), pp. 55–68

    Google Scholar 

  14. R. Eldan, J. Lehec, Bounding the norm of a log-concave vector via thin-shell estimates, in Geometric Aspects of Functional Analysis, ed. by B. Klartag, E. Milman. Lecture Notes in Mathematics, vol. 2116 (Springer, Berlin, 2014), pp. 107–122

    Google Scholar 

  15. M. Fradelizi, Sections of convex bodies through their centroid. Arch. Math. (Basel) 69 (6), 515–522 (1997)

    Google Scholar 

  16. M. Fradelizi, M. Meyer, Increasing functions and inverse Santaló inequality for unconditional functions. Positivity 12 (3), 407–420 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  17. J. Li, M. Fradelizi, M. Madiman, Concentration of information content and other functionals under convex measures, in Proceedings of the 2016 IEEE International Symposium on Information Theory (Barcelona, 2016) pp. 1128–1132

    Google Scholar 

  18. O. Guédon, E. Milman, Interpolating thin-shell and sharp large-deviation estimates for isotropic log-concave measures. Geom. Funct. Anal. 21 (5), 1043–1068 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  19. G. Hargé, Reinforcement of an inequality due to Brascamp and Lieb. J. Funct. Anal. 254 (2), 267–300 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. J.-B. Hiriart-Urruty, C. Lemaréchal, Convex Analysis and Minimization Algorithms. I. Fundamentals. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 305 (Springer-Verlag, Berlin, 1993)

    Google Scholar 

  21. B. Klartag, A central limit theorem for convex sets. Invent. Math. 168 (1), 91–131 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  22. B. Klartag, Eigenvalue distribution of optimal transportation. in Presentation given at the Workshop on Information Theory and Concentration Phenomena held at the Institute for Mathematics and its Applications (IMA), Minneapolis, April (2015)

    Google Scholar 

  23. B. Klartag, A. Kolesnikov, Eigenvalue distribution of optimal transportation. Anal. PDE 8 (1), 33–55 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  24. B. Klartag, V.D. Milman, Geometry of log-concave functions and measures. Geom. Dedicata 112, 169–182 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  25. E.H. Lieb, M. Loss, Analysis, 2nd edn. Graduate Studies in Mathematics, vol. 14 (American Mathematical Society, Providence, RI, 2001)

    Google Scholar 

  26. M. Madiman, L. Wang, S. Bobkov, Some applications of the nonasymptotic equipartition property of log-concave distributions. Preprint (2016)

    Google Scholar 

  27. V.H. Nguyen, Inégalités fonctionnelles et convexité. Ph.D. thesis, Université Pierre et Marie Curie (Paris VI) (2013)

    Google Scholar 

  28. V.H. Nguyen, Dimensional variance inequalities of Brascamp-Lieb type and a local approach to dimensional Prékopa’s theorem. J. Funct. Anal. 266 (2), 931–955 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  29. A. Prékopa, On logarithmic concave measures and functions. Acta Sci. Math. (Szeged) 34, 335–343 (1973)

    MathSciNet  MATH  Google Scholar 

  30. R.T. Rockafellar, Convex Analysis. Princeton Mathematical Series, vol. 28 (Princeton University Press, Princeton, NJ, 1970)

    Google Scholar 

  31. A. van der Vaart, J.A. Wellner, A local maximal inequality under uniform entropy. Electron. J. Stat. 5, 192–203 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  32. L. Wang, Heat capacity bound, energy fluctuations and convexity. Ph.D. thesis, Yale University (2014)

    Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  34. J.A. Wellner, Limit theorems for the ratio of the empirical distribution function to the true distribution function. Z. Wahrsch. Verw. Gebiete 45 (1), 73–88 (1978)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are indebted to Paata Ivanisvili, Fedor Nazarov, and Christos Saroglou for useful comments on an earlier version of this paper. In particular, Christos Saroglou pointed out that the class of extremals in our inequalities is wider than we had realized, and Remark 2.5 is due to him. We are also grateful to François Bolley, Dario Cordero-Erausquin and an anonymous referee for pointing out relevant references.

This work was partially supported by the project GeMeCoD ANR 2011 BS01 007 01, and by the U.S. National Science Foundation through the grant DMS-1409504 (CAREER). A significant portion of this paper is based on the Ph.D. dissertation of Wang [32], co-advised by M. Madiman and N. Read, at Yale University

Author information

Authors and Affiliations

  1. Laboratoire d’Analyse et de Mathématiques Appliquées UMR 8050, Université Paris-Est Marne-la-Vallée, 5 Bd Descartes, Champs-sur-Marne, 77454, Marne-la-Vallée Cedex 2, France

    Matthieu Fradelizi

  2. Department of Mathematical Sciences, University of Delaware, 501 Ewing Hall, Newark, DE, 19716, USA

    Mokshay Madiman

  3. J.P. Morgan, New York, NY, USA

    Liyao Wang

Authors
  1. Matthieu Fradelizi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Mokshay Madiman
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Liyao Wang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mokshay Madiman .

Editor information

Editors and Affiliations

  1. Georgia Institute of Technology, Atlanta, Georgia, USA

    Christian Houdré

  2. University of Delaware, Department of Applied Economics and Stat University of Delaware, Newark, Delaware, USA

    David M. Mason

  3. Centre national de la recherche scientifique, Laboratoire J.A. Dieudonné, Université Côte d’Azur, Nice, France

    Patricia Reynaud-Bouret

  4. Department of Mathematics, University of Tennessee Department of Mathematics, Knoxville, Tennessee, USA

    Jan Rosiński

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this paper

Cite this paper

Fradelizi, M., Madiman, M., Wang, L. (2016). Optimal Concentration of Information Content for Log-Concave Densities. In: Houdré, C., Mason, D., Reynaud-Bouret, P., Rosiński, J. (eds) High Dimensional Probability VII. Progress in Probability, vol 71. Birkhäuser, Cham. https://doi.org/10.1007/978-3-319-40519-3_3

Download citation

Publish with us

Policies and ethics

Search

Navigation