Optimal Coreset for Gaussian Kernel Density Estimation

Abstract: Given a point set $P\subset \mathbb{R}<sup>d$, the kernel density estimate of $P$ is defined as [ \overline{\mathcal{G}}P(x) = \frac{1}{\left|P\right|}\sum{p\in P}e^{-\left\lVert x-p \right\rVert^2} ] for any $x\in\mathbb{R}<sup>d$. We study how to construct a small subset $Q$ of $P$ such that the kernel density estimate of $P$ is approximated by the kernel density estimate of $Q$. This subset $Q$ is called a coreset. The main technique in this work is constructing a $\pm 1$ coloring on the point set $P$ by discrepancy theory and we leverage Banaszczyk's Theorem. When $d&gt;1$ is a constant, our construction gives a coreset of size $O\left(\frac{1}{\varepsilon}\right)$ as opposed to the best-known result of $$O\left(\frac{1}{\varepsilon}\sqrt{\log\frac{1}{\varepsilon}}\right)$$. It is the first result to give a breakthrough on the barrier of $\sqrt{\log}$ factor even when $d=2$.