Optimal Quantisation of Probability Measures Using Maximum Mean Discrepancy

Several researchers have proposed minimisation of maximum mean discrepancy (MMD) as a method to quantise probability measures, i.e., to approximate a target distribution by a representative point set. Here we consider sequential algorithms that greedily minimise MMD over a discrete candidate set. We propose a novel non-myopic algorithm and, in order to both improve statistical efficiency and reduce computational cost, we investigate a variant that applies this technique to a mini-batch of the candidate set at each iteration. When the candidate points are sampled from the target, the consistency of these new algorithm - and their mini-batch variants - is established. We demonstrate the algorithms on a range of important computational problems, including optimisation of nodes in Bayesian cubature and the thinning of Markov chain output.