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 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*(n4) and in amortized time O*(n3) 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.
ASJC Scopus subject areas
- Computer Graphics and Computer-Aided Design
- Applied Mathematics