Hashing-Based-Estimators for Kernel Density in High Dimensions

Given a set of points P⊄ R^d and a kernel k, the Kernel Density Estimate at a point x∊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.