Optimal Coreset for Gaussian Kernel Density Estimation
This provides a breakthrough for computational geometry and machine learning applications by reducing the barrier of a √log factor in coreset construction, even in low dimensions like d=2.
The paper tackles the problem of constructing small coresets for Gaussian kernel density estimation, achieving a coreset size of O(1/ε) for constant dimensions, which improves upon the previous best-known result of O(1/ε √log(1/ε)).
Given a point set $P\subset \mathbb{R}^d$, 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 $x\in\mathbb{R}^d$. 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 $d>1$ 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 $d=2$.