Distribution Compression in Near-linear Time

In distribution compression, one aims to accurately summarize a probability distribution mathbbP using a small number of representative points. Near-optimal thinning procedures achieve this goal by sampling n points from a Markov chain and identifying √n points with widetildeO(1/√n) discrepancy to mathbbP. Unfortunately, these algorithms suffer from quadratic or super-quadratic runtime in the sample size n. To address this deficiency, we introduce Compress++, a simple meta-procedure for speeding up any thinning algorithm while suffering at most a factor of 4 in error. When combined with the quadratic-time kernel halving and kernel thinning algorithms of Dwivedi and Mackey (2021), Compress++ delivers √n points with O(√log n/n) integration error and better-than-Monte-Carlo maximum mean discrepancy in O(n log3 n) time and O( √n log2 n ) space. Moreover, Compress++ enjoys the same near-linear runtime given any quadratic-time input and reduces the runtime of super-quadratic algorithms by a square-root factor. In our benchmarks with high-dimensional Monte Carlo samples and Markov chains targeting challenging differential equation posteriors, Compress++ matches or nearly matches the accuracy of its input algorithm in orders of magnitude less time.