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 \(\sqrt{n}\) points with \(Õ(1/\sqrt{n})\) 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 \(\sqrt{n}\) points with \(O(\sqrt{log n/n})\) integration error and better-than-Monte-Carlo maximum mean discrepancy in \(O(\sqrt{n log^3 n})\) time and \(O(\sqrt{n}log^2 n)\) 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.