Stronger Coreset Bounds for Kernel Density Estimators via Chaining
Rainie Bozzai Thomas Rothvoss
We apply the discrepancy method and a chaining approach to give improved
bounds on the coreset complexity of a wide class of kernel functions. Our
results give randomized polynomial time algorithms to produce coresets of size
$O\big(\frac{\sqrt{d}}{\varepsilon}\sqrt{\log\log \frac{1}{\varepsilon}}\big)$
for the Gaussian and Laplacian kernels in the case that the data set is
uniformly bounded, an improvement that was not possible with previous
techniques. We also obtain coresets of size
$O\big(\frac{1}{\varepsilon}\sqrt{\log\log \frac{1}{\varepsilon}}\big)$ for the
Laplacian kernel for $d$ constant. Finally, we give the best known bounds of
$O\big(\frac{\sqrt{d}}{\varepsilon}\sqrt{\log(2\max\{1,\alpha\})}\big)$ on the
coreset complexity of the exponential, Hellinger, and JS Kernels, where
$1/\alpha$ is the bandwidth parameter of the kernel.