Skip to main content
SpringerLink
Log in

Gibbs/Metropolis algorithms on a convex polytope

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract

This paper gives sharp rates of convergence for natural versions of the Metropolis algorithm for sampling from the uniform distribution on a convex polytope. The singular proposal distribution, based on a walk moving locally in one of a fixed, finite set of directions, needs some new tools. We get useful bounds on the spectrum and eigenfunctions using Nash and Weyl-type inequalities. The top eigenvalues of the Markov chain are closely related to the Neumann eigenvalues of the polytope for a novel Laplacian.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Chatterjee, S., Diaconis, P., Sly, A.: Properties of uniform doubly stochastic matrices. ArXiv e-prints (2010)

  2. Diaconis P., Anderson H.C.: Hit and run as a unifying device. J. Soc. Francaise Stat. 148, 5–28 (2007)

    Google Scholar 

  3. Diaconis P., Lebeau G., Michel L.: Geometric analysis for the Metropolis algorithm on Lipshitz domains. Invent. Math. 185(2), 239–281 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Diaconis, P., Matchup-Wood, P.: On random, doubly stochastic, tri-diagonal matrices. Technical report, Department of Mathematics, Stanford University, preprint, (2010)

  5. Diaconis P., Saloff-Coste L.: do we know about the Metropolis algorithm? In: 27th Annual ACM Symposium on the Theory of Computing (STOC’95) (Las Vegas, NV). J. Comput. Syst. Sci. 57(1), 20–36 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  6. Diaconis P., Lebeau G.: Micro-local analysis for the Metropolis algorithm. Math. Z. 262(2), 411–447 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  7. Lebeau G., Michel L.: Semiclassical analysis of a random walk on a manifold. Ann. Probab. 38(1), 277–315 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  8. Liu J.S.: Monte Carlo Strategies in Scientific Computing. Springer Series in Statistics. Springer-Verlag, New York (2001)

    Google Scholar 

  9. Lovász L., Simonovits M.: Random walks in a convex body and an improved volume algorithm. Random Struct. Algorithms 4(4), 359–412 (1993)

    Article  MATH  Google Scholar 

  10. Lovász, L., Vempala, S.: Hit-and-run is fast and fun. Technical Report, Microsoft Research, Microsoft Corporation (2003)

  11. Lovász L., Vempala S.: The geometry of logconcave functions and sampling algorithms. Random Struct. Algorithms 30(3), 307–358 (2007)

    Article  MATH  Google Scholar 

  12. Meyn S.P., Tweedie R.L.: Markov Chains and Stochastic Stability. Communications and Control Engineering Series. Springer-Verlag, London (1993)

    Book  Google Scholar 

  13. Reed M., Simon B.: Methods of Modern Mathematical Physics. IV. Analysis of Operators. Academic Press/Harcourt Brace Jovanovich Publishers, New York (1978)

    MATH  Google Scholar 

  14. Rosenthal J.S.: Minorization conditions and convergence rates for Markov chain Monte Carlo. J. Am. Stat. Assoc. 90(430), 558–566 (1995)

    Article  MATH  Google Scholar 

  15. Smith R.L.: Efficient Monte Carlo procedures for generating points uniformly distributed over bounded regions. Oper. Res. 32(6), 1296–1308 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  16. Yuen W.K.: Applications of geometric bounds to the convergence rate of Markov chains on \({{\mathbb{R}^n}}\) . Stoch. Process. Appl. 87(1), 1–23 (2000)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Departments of Mathematics and Statistics, Stanford University, Stanford, USA

    Persi Diaconis, Gilles Lebeau & Laurent Michel

  2. Département de Mathématiques, Parc Valrose, Université de Nice Sophia-Antipolis, 06108, Nice Cedex 02, France

    Persi Diaconis, Gilles Lebeau & Laurent Michel

Authors
  1. Persi Diaconis
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Gilles Lebeau
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Laurent Michel
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Gilles Lebeau.

Additional information

Persi Diaconis: Supported in part by NSF grant 0804324.

Gilles Lebeau: Supported in part by ANR-06-BLAN-0250-03.

Laurent Michel: Supported in part by ANR-06-BLAN-0250-03 and ANR-09-JCJC-0099-01.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Diaconis, P., Lebeau, G. & Michel, L. Gibbs/Metropolis algorithms on a convex polytope. Math. Z. 272, 109–129 (2012). https://doi.org/10.1007/s00209-011-0924-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-011-0924-5

Keywords

Search

Navigation