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.