Stochastic billiards for sampling from the boundary of a convex set

Stochastic billiards can be used for approximate sampling from the boundary of a bounded convex set through the Markov Chain Monte Carlo (MCMC) paradigm. This paper studies how many steps of the underlying Markov chain are required to get samples (approximately) from the uniform distribution on the boundary of the set, for sets with an upper bound on the curvature of the boundary. Our main theorem implies a polynomial-time algorithm for sampling from the boundary of such sets

Closures of exponential families

The variation distance closure of an exponential family with a convex set of canonical parameters is described, assuming no regularity conditions. The tools are the concepts of convex core of a measure and extension of an exponential family, introduced previously by the authors, and a new concept of accessible faces of a convex set. Two other closures related to the information divergence are also characterized.Comment: Published at http://dx.doi.org/10.1214/009117904000000766 in the Annals of Probability (http://www.imstat.org/aop/) by the Institute of Mathematical Statistics (http://www.imstat.org

Minimum L1-distance projection onto the boundary of a convex set: Simple characterization

We show that the minimum distance projection in the L1-norm from an interior point onto the boundary of a convex set is achieved by a single, unidimensional projection. Application of this characterization when the convex set is a polyhedron leads to either an elementary minmax problem or a set of easily solved linear programs, depending upon whether the polyhedron is given as the intersection of a set of half spaces or as the convex hull of a set of extreme points. The outcome is an easier and more straightforward derivation of the special case results given in a recent paper by Briec.Comment: 5 page

A Cubic Algorithm for Computing Gaussian Volume

We present randomized algorithms for sampling the standard Gaussian distribution restricted to a convex set and for estimating the Gaussian measure of a convex set, in the general membership oracle model. The complexity of integration is $O^*(n^3)$ while the complexity of sampling is $O^*(n^3)$ for the first sample and $O^*(n^2)$ for every subsequent sample. These bounds improve on the corresponding state-of-the-art by a factor of $n$. Our improvement comes from several aspects: better isoperimetry, smoother annealing, avoiding transformation to isotropic position and the use of the "speedy walk" in the analysis.Comment: 23 page