Distribution Compression in Near-linear Time
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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 𝒪(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^3 n) time and 𝒪( √(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.
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