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 KDE _P (x) = 1/|P| Σy _∈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.