Kernel Thinning

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 $O(n<sup>2)$ 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 $O_d(n^{{-1/2}\sqrt{\log} n})$ 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 $\mathbb{R}<sup>d$. In contrast, an equal-sized i.i.d. sample from $\mathbb{P}$ suffers $\Omega(n<sup>{-1/4})$ integration error. Our sub-exponential guarantees resemble the classical quasi-Monte Carlo error rates for uniform $\mathbb{P}$ on $[0,1]<sup>d$ but apply to general distributions on $\mathbb{R}<sup>d$ and a wide range of common kernels. Moreover, the same construction delivers near-optimal $L<sup>\infty$ coresets in $O(n<sup>2)$ 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 $100$.