### Abstract

The class of logconcave functions in ℝ^{n} 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* (n^{4}), and in amortized time O* (n^{3}) 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 language | English |
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Title of host publication | Annual Symposium on Foundations of Computer Science - Proceedings |

Pages | 640-649 |

Number of pages | 10 |

Publication status | Published - 2003 |

Event | Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003 - Cambridge, MA, United States Duration: Oct 11 2003 → Oct 14 2003 |

### Other

Other | Proceedings: 44th Annual IEEE Symposium on Foundations of Computer Science - FOCS 2003 |
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Country | United States |

City | Cambridge, MA |

Period | 10/11/03 → 10/14/03 |

### Fingerprint

### ASJC Scopus subject areas

- Hardware and Architecture

### Cite this

*Annual Symposium on Foundations of Computer Science - Proceedings*(pp. 640-649)

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

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*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.

}

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 -