Efficient Multiscale Sampling from Products of Gaussian Mixtures

The problem of approximating the product of several Gaussian mixture distributions arises in a number of contexts, including the nonparametric belief propagation (NBP) inference algorithm and the training of product of experts models. This paper develops two multiscale algorithms for sampling from a product of Gaussian mixtures, and compares their performance to existing methods. The first is a multiscale variant of previously proposed Monte Carlo techniques, with comparable theoretical guarantees but improved empirical convergence rates. The second makes use of approximate kernel density evaluation methods to construct a fast approximate sampler, which is guaranteed to sample points to within a tunable parameter ɛ of their true probability. We compare both multiscale samplers on a set of computational examples motivated by NBP, demonstrating significant improvements over existing methods.