### 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 | Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS |

Publisher | IEEE Computer Society |

Pages | 640-649 |

Number of pages | 10 |

Volume | 2003-January |

ISBN (Print) | 0769520405 |

DOIs | |

Publication status | Published - 2003 |

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

### Other

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

City | Cambridge |

Period | 10/11/03 → 10/14/03 |

### Fingerprint

### 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

*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.

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

*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

}

TY - GEN

T1 - Logconcave functions

T2 - Geometry and efficient sampling algorithms

AU - Lovász, L.

AU - Vempala, S.

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.

KW - Density functional theory

KW - Engineering profession

KW - Filters

KW - Gaussian processes

KW - Geometry

KW - Lattices

KW - Mathematics

KW - Probability distribution

KW - Sampling methods

KW - Stochastic processes

UR - http://www.scopus.com/inward/record.url?scp=84858543840&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84858543840&partnerID=8YFLogxK

U2 - 10.1109/SFCS.2003.1238236

DO - 10.1109/SFCS.2003.1238236

M3 - Conference contribution

SN - 0769520405

VL - 2003-January

SP - 640

EP - 649

BT - Proceedings - Annual IEEE Symposium on Foundations of Computer Science, FOCS

PB - IEEE Computer Society

ER -