Logconcave functions

Geometry and efficient sampling algorithms

L. Lovász, Santosh Vempala

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

3 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 publicationAnnual Symposium on Foundations of Computer Science - Proceedings
Pages640-649
Number of pages10
Publication statusPublished - 2003
EventProceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003 - Cambridge, MA, United States
Duration: Oct 11 2003Oct 14 2003

Other

OtherProceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003
CountryUnited States
CityCambridge, MA
Period10/11/0310/14/03

Fingerprint

Probability density function
Sampling
Geometry

ASJC Scopus subject areas

  • Hardware and Architecture

Cite this

Lovász, L., & Vempala, S. (2003). Logconcave functions: Geometry and efficient sampling algorithms. In Annual Symposium on Foundations of Computer Science - Proceedings (pp. 640-649)

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

Annual Symposium on Foundations of Computer Science - Proceedings. 2003. p. 640-649.

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

Lovász, L & Vempala, S 2003, Logconcave functions: Geometry and efficient sampling algorithms. in Annual Symposium on Foundations of Computer Science - Proceedings. pp. 640-649, Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003, Cambridge, MA, United States, 10/11/03.
Lovász L, Vempala S. Logconcave functions: Geometry and efficient sampling algorithms. In Annual Symposium on Foundations of Computer Science - Proceedings. 2003. p. 640-649
Lovász, L. ; Vempala, Santosh. / Logconcave functions : Geometry and efficient sampling algorithms. Annual Symposium on Foundations of Computer Science - Proceedings. 2003. pp. 640-649
@inproceedings{d1b91e7a7e76434ca55044e4ce2e1194,
title = "Logconcave functions: Geometry and efficient sampling algorithms",
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.",
author = "L. Lov{\'a}sz and Santosh Vempala",
year = "2003",
language = "English",
pages = "640--649",
booktitle = "Annual Symposium on Foundations of Computer Science - Proceedings",

}

TY - GEN

T1 - Logconcave functions

T2 - Geometry and efficient sampling algorithms

AU - Lovász, L.

AU - Vempala, Santosh

PY - 2003

Y1 - 2003

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=17444438746&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=17444438746&partnerID=8YFLogxK

M3 - Conference contribution

SP - 640

EP - 649

BT - Annual Symposium on Foundations of Computer Science - Proceedings

ER -