Kernel Thinning

• URL: http://arxiv.org/abs/2105.05842v9
• Date: Wed, 7 Jun 2023 00:41:12 GMT
• Title: Kernel Thinning
• Authors: Raaz Dwivedi, Lester Mackey
• Abstract summary: kernel thinning is a new procedure for compressing a distribution $mathbbP$ more effectively than i.i.d. sampling or standard thinning. We derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat'ern, and B-spline kernels.
• Score: 26.25415159542831
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: We introduce kernel thinning, a new procedure for compressing a distribution $\mathbb{P}$ more effectively than i.i.d. sampling or standard thinning. Given a suitable reproducing kernel $\mathbf{k}_{\star}$ and $\mathcal{O}(n^2)$ time, kernel thinning compresses an $n$-point approximation to $\mathbb{P}$ into a $\sqrt{n}$-point approximation with comparable worst-case integration error across the associated reproducing kernel Hilbert space. The maximum discrepancy in integration error is $\mathcal{O}_d(n^{-1/2}\sqrt{\log n})$ in probability for compactly supported $\mathbb{P}$ and $\mathcal{O}_d(n^{-\frac{1}{2}} (\log n)^{(d+1)/2}\sqrt{\log\log n})$ for sub-exponential $\mathbb{P}$ on $\mathbb{R}^d$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n^{-1/4})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]^d$ but apply to general distributions on $\mathbb{R}^d$ and a wide range of common kernels. Moreover, the same construction delivers near-optimal $L^\infty$ coresets in $\mathcal O(n^2)$ time. We use our results to derive explicit non-asymptotic maximum mean discrepancy bounds for Gaussian, Mat\'ern, and B-spline kernels and present two vignettes illustrating the practical benefits of kernel thinning over i.i.d. sampling and standard Markov chain Monte Carlo thinning, in dimensions $d=2$ through $100$.

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