Hashing-Based-Estimators for Kernel Density in High Dimensions

Abstract

Given a set of points $P\subset {\mathbb{R}}^d$ and a kernel $k$, the Kernel Density Estimate at a point $x\in {\mathbb{R}}^d$ is defined as ${\mathrm{KDE}}_P(x)=\frac{1}{|P|}{\sum }_{y\in P}k(x,y)$. We study the problem of designing a data structure that given a data set $P$ and a kernel function, returns *approximations to the kernel density* of a query point in *sublinear time*. We introduce a class of unbiased estimators for kernel density implemented through locality-sensitive hashing, and give general theorems bounding the variance of such estimators. These estimators give rise to efficient data structures for estimating the kernel density in high dimensions for a variety of commonly used kernels. Our work is the first to provide data-structures with theoretical guarantees that improve upon simple random sampling in high dimensions.

A preliminary version of this paper appeared in FOCS 2017