Hashing-Based-Estimators for Kernel Density in High Dimensions

08/30/2018

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by   Moses Charikar, et al.

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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|∑_y∈ 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.