Improved Coresets for Kernel Density Estimates

TL;DR

This paper develops improved methods for constructing small coresets that accurately approximate kernel density estimates, with tight bounds and analysis applicable across various kernels and dimensions, enhancing efficiency in statistical and geometric applications.

Contribution

It introduces new bounds and analysis for coresets approximating kernel density estimates, especially for characteristic kernels, and refines bounds for Gaussian kernels in fixed dimensions.

Findings

Coreset size is $2/\epsilon^2$ for characteristic kernels, independent of data dimension.

Tighter bounds are achieved for Gaussian kernels when the dimension is constant.

The analysis unifies approaches from statistics, machine learning, and geometry.

Abstract

We study the construction of coresets for kernel density estimates. That is we show how to approximate the kernel density estimate described by a large point set with another kernel density estimate with a much smaller point set. For characteristic kernels (including Gaussian and Laplace kernels), our approximation preserves the $L_\infty$ error between kernel density estimates within error $\epsilon$, with coreset size $2/\epsilon^2$, but no other aspects of the data, including the dimension, the diameter of the point set, or the bandwidth of the kernel common to other approximations. When the dimension is unrestricted, we show this bound is tight for these kernels as well as a much broader set. This work provides a careful analysis of the iterative Frank-Wolfe algorithm adapted to this context, an algorithm called \emph{kernel herding}. This analysis unites a broad line of work that spans statistics, machine learning, and geometry. When the dimension $d$ is constant, we demonstrate much tighter bounds on the size of the coreset specifically for Gaussian kernels, showing that it is bounded by the size of the coreset for axis-aligned rectangles. Currently the best known constructive bound is $O(\frac{1}{\epsilon} \log^d \frac{1}{\epsilon})$, and non-constructively, this can be improved by $\sqrt{\log \frac{1}{\epsilon}}$. This improves the best constant dimension bounds polynomially for $d \geq 3$.