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 - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003
PublisherIEEE Computer Society
Pages640-649
Number of pages10
ISBN (Electronic)0769520405
DOIs
Publication statusPublished - Jan 1 2003
Event44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 - Cambridge, United States
Duration: Oct 11 2003Oct 14 2003

Publication series

NameProceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS
Volume2003-January
ISSN (Print)0272-5428

Other

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

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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 - 44th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2003 (pp. 640-649). [1238236] (Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS; Vol. 2003-January). IEEE Computer Society. https://doi.org/10.1109/SFCS.2003.1238236