  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 $O(\sqrt{d}/\varepsilon\cdot
\sqrt{\log 1/\varepsilon} )$, and we show a near-matching lower bound of size
$\Omega(\min\{\sqrt{d}/\varepsilon, 1/\varepsilon^2\})$. When $d\geq
1/\varepsilon^2$, 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.
