Distribution Compression in Near-linear Time

In distribution compression, one aims to accurately summarize a probability distribution P 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 Õ(1/ √ n) discrepancy to P. 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 log n) time and O( √ n log 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.