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 \\(\u00d5(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.<\/p>\n","protected":false},"excerpt":{"rendered":"
In distribution compression, one aims to accurately summarize a probability distribution using a small number of representative points. Near-optimal thinning procedures achieve this goal by sampling points from a Markov chain and identifying points with discrepancy to . Unfortunately, these algorithms suffer from quadratic or super-quadratic runtime in the sample size . To address this 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