### 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 logconcave density functions, we study their geometry and introduce a technique for "smoothing" them out. These results are applied to analyze two efficient algorithms for sampling from a logconcave distribution in n dimensions, with no assumptions on the local smoothness of the density function. Both algorithms, the ball walk and the hit-and-run walk, use a random walk (Markov chain) to generate a random point. After appropriate preprocessing, they produce a point from approximately the right distribution in time O*(n^{4}) and in amortized time O*(n^{3}) if n or more sample points are needed (where the asterisk indicates that dependence on the error parameter and factors of log n are not shown). These bounds match previous bounds for the special case of sampling from the uniform distribution over a convex body.

Original language | English |
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Pages (from-to) | 307-358 |

Number of pages | 52 |

Journal | Random Structures and Algorithms |

Volume | 30 |

Issue number | 3 |

DOIs | |

Publication status | Published - May 1 2007 |

### ASJC Scopus subject areas

- Software
- Mathematics(all)
- Computer Graphics and Computer-Aided Design
- Applied Mathematics

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## Cite this

*Random Structures and Algorithms*,

*30*(3), 307-358. https://doi.org/10.1002/rsa.20135