Sampling and integration of near log-concave functions

TLDR: This work provides the first polynomial time algorithm to generate samples from a given log-concave distribution and proves a general isoperimetric inequality for convex sets and uses this together with recent developments in the theory of rapidly mixing Markov chains.

Abstract: Sampling and Integration of Near Log-Concave Functions David Applegate* Ravi Kannan* An important class of functions that arise in statistics and other areas are the log-concave functions. Here we provide the first polynomial time algorithm to generate samples from a given log-concave distribution. The algorithm is fairly simple and natural; it is the proof of its fast convergence that is new. To this end, we prove a general isoperimetric inequality for convex sets and use this together with recent developments in the theory of rapidly mixing Markov chains. We use our sampling algorithm to develop an algorithm for integrating log-concave functions. As one application, we are able to develop an algorithm for approximating the volume of convex bodies given by an oracle; we do so by enclosing the given body in a cube, defining a log-concave function that is 1 on the body and exponentially falls off outside, and integrating this function. This a.llo ws us to avoid one of the complications in prior algorithms for computing volumes – dealing with sharp corners – and results in an algorithm which is faster than previous algorithms.