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arXiv · 1808.10530

Hashing-Based-Estimators for Kernel Density in High Dimensions

Moses Charikar·Paris Siminelakis
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Abstract

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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Moses Charikar, Paris Siminelakis. 2018-08-30. Hashing-Based-Estimators for Kernel Density in High Dimensions. https://arxiv.org/abs/1808.10530

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