Kernel Thinning
(2105.05842)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 $O(n2)$ 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 $Od(n{-1/2}\sqrt{\log n})$ in probability for compactly supported $\mathbb{P}$ and $O_d(n{-\frac{1}{2}} (\log n){(d+1)/2}\sqrt{\log\log n})$ 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 $O(n2)$ 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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