Optimal quantisation of probability measures using maximum mean discrepancy
This work addresses computational problems like Bayesian cubature and Markov chain thinning, offering incremental improvements in efficiency for researchers in statistics and machine learning.
The authors tackled the problem of quantising probability measures by approximating a target distribution with a representative point set, proposing a novel non-myopic algorithm and a mini-batch variant that improve efficiency and reduce computational cost, with consistency proven when candidate points are sampled from the target.
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. 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.