Quantization of Measures via Interacting Particle-based Optimization

We will discuss several interacting particle systems for sampling, or more precisely quantization of target measures.  Namely instead of seeking an i.i.d. sample of the target measure we look to approximate the target probability distribution by a family of particles.

This  can be cast as an optimization problem  where the objective functional measures the dissimilarity to the target. This optimization can be addressed by approximating  Wasserstein and related gradient flows. We will compare and contrast the Stein Variational Gradient Descent, projected  gradient flows and gradient flows of  Maximum Mean and  Kernel Stein Discrepancy.

In practice, these are simulated by interacting particle systems, whose stationary states define an empirical measure approximating the target distribution.   We investigate,  theoretically and numerically, quantization properties  of these approaches, i.e. how well is the target approximated by the empirical measure. In particular, we will discuss  upper bounds  on the quantization error of MMD and KSD with various kernels.   The talk is based on joint work with Anna Korba and Lantian Xu.