Near-Optimal Coresets of Kernel Density Estimates

We construct near-optimal coresets for kernel density estimates for points in $mathbb{R}^d$ when the kernel is positive definite. Specifically we show a polynomial time construction for a coreset of size $O(sqrt{d}/varepsiloncdot sqrt{log 1/varepsilon} )$, and we show a near-matching lower bound of size $Omega(min{sqrt{d}/varepsilon, 1/varepsilon^2})$. When $dgeq 1/varepsilon^2$, it is known that the size of coreset can be $O(1/varepsilon^2)$. The upper bound is a polynomial-in-$(1/varepsilon)$ improvement when $d in [3,1/varepsilon^2)$ and the lower bound is the first known lower bound to depend on $d$ for this problem. Moreover, the upper bound restriction that the kernel is positive definite is significant in that it applies to a wide-variety of kernels, specifically those most important for machine learning. This includes kernels for information distances and the sinc kernel which can be negative.