Logconcave functions

Geometry and efficient sampling algorithms

L. Lovász, S. Vempala

Research output: Chapter in Book/Report/Conference proceedingConference contribution

7 Citations (Scopus)

Abstract

The class of logconcave functions in ℝn is a common generalization of Gaussians and of indicator functions of convex sets. Motivated by the problem of sampling from a logconcave density function, we study their geometry and introduce an analysis technique for "smoothing" them out. This leads to efficient sampling algorithms with no assumptions on the local smoothness of the density function. After appropriate preprocessing, both the ball walk (with a Metropolis filter) and a generalization of hit-and-run produce a point from approximately the right distribution in time O∗(n4), and in amortized time O∗(n3) if many sample points are needed (where the asterisk indicates that dependence on the error parameter and factors of log n are not shown). The bounds are optimal in terms of a "roundness" parameter and match the best-known bounds for the special case of the uniform density over a convex set.

Original languageEnglish
Title of host publicationProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
PublisherIEEE Computer Society
Pages640-649
Number of pages10
Volume2003-January
ISBN (Print)0769520405
DOIs
Publication statusPublished - 2003
Event44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States
Duration: Oct 11 2003Oct 14 2003

Other

Other44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
CountryUnited States
CityCambridge
Period10/11/0310/14/03

Fingerprint

Probability density function
Sampling
Geometry

Keywords

  • Density functional theory
  • Engineering profession
  • Filters
  • Gaussian processes
  • Geometry
  • Lattices
  • Mathematics
  • Probability distribution
  • Sampling methods
  • Stochastic processes

ASJC Scopus subject areas

  • Computer Science(all)

Cite this

Lovász, L., & Vempala, S. (2003). Logconcave functions: Geometry and efficient sampling algorithms. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS (Vol. 2003-January, pp. 640-649). [1238236] IEEE Computer Society. https://doi.org/10.1109/SFCS.2003.1238236

Logconcave functions : Geometry and efficient sampling algorithms. / Lovász, L.; Vempala, S.

Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. Vol. 2003-January IEEE Computer Society, 2003. p. 640-649 1238236.

Research output: Chapter in Book/Report/Conference proceedingConference contribution

Lovász, L & Vempala, S 2003, Logconcave functions: Geometry and efficient sampling algorithms. in Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. vol. 2003-January, 1238236, IEEE Computer Society, pp. 640-649, 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003, Cambridge, United States, 10/11/03. https://doi.org/10.1109/SFCS.2003.1238236
Lovász L, Vempala S. Logconcave functions: Geometry and efficient sampling algorithms. In Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. Vol. 2003-January. IEEE Computer Society. 2003. p. 640-649. 1238236 https://doi.org/10.1109/SFCS.2003.1238236
Lovász, L. ; Vempala, S. / Logconcave functions : Geometry and efficient sampling algorithms. Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS. Vol. 2003-January IEEE Computer Society, 2003. pp. 640-649
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