Given a point set P\subset R^d,  kernel density estimation for Gaussian kernel is defined as \mathcal{G}_P(x) = 1{\P\}p\in Pe^{-\left\lVert x-p \right\rVert^2} for any x\inR^d. We study how to construct a small subset Q of P such that the kernel density estimation of P can be approximated by the kernel density estimation of Q. This subset Q is called coreset . The primary technique in this work is to construct \pm 1 coloring on the point set P by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyk's Theorem. When d>1 is constant, our construction gives a coreset of size O\left(1{\varepsilon} \sqrt{\log\log\frac{1{\varepsilon}}} \right) as opposed to the best-known result of O\left(1{\varepsilon}\log\frac{1{\varepsilon}}\right). It is the first to give a breakthrough on the barrier of \log factor even when d=2.
Given a point set P\subset R^d, a kernel density estimation for Gaussian kernel is defined as \mathcal{G}_P(x) = 1{\P\}p\in Pe^{-\left\lVert x-p \right\rVert^2} for any x\inR^d. We study how to construct a small subset Q of P such that the kernel density estimation of P can be approximated by the kernel density estimation of Q. This subset Q is called coreset . The primary technique in this work is to construct \pm 1 coloring on the point set P by the discrepancy theory and apply this coloring algorithm recursively. Our result leverages Banaszczyk's Theorem. When d>1 is constant, our construction gives a coreset of size O\left(1{\varepsilon} {\varepsilon}}} \right) as opposed to the best-known result of O\left(1{\varepsilon}\log\frac{1{\varepsilon}}\right). It is the first to give a breakthrough on the barrier of \log factor even when d=2.
A
a
R
coreset
coreset
D
\sqrt{\log\log\frac{1