Hashing-based-estimators For Kernel Density In High Dimensions

Charikar Moses, Siminelakis Paris. Arxiv 2018

[Paper]    
ARXIV Independent

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.

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