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Given a set of points $P\subset \mathbb{R}^{d}$ and a kernel $k$, the KernelDensity 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 problemof 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 *sublineartime*. We introduce a class of unbiased estimators for kernel densityimplemented through locality-sensitive hashing, and give general theoremsbounding the variance of such estimators. These estimators give rise toefficient data structures for estimating the kernel density in high dimensionsfor a variety of commonly used kernels. Our work is the first to providedata-structures with theoretical guarantees that improve upon simple randomsampling in high dimensions.

Hashing-Based-Estimators for Kernel Density in High Dimensions