Kernel Density Estimation through Density Constrained Near Neighbor Search

TL;DR

This paper introduces a new data structure for efficient kernel density estimation in high-dimensional spaces, improving query time and space complexity by leveraging geometric structure and data-dependent near neighbor search techniques.

Contribution

It presents a novel implementation of importance sampling combined with geometric analysis to enhance kernel density estimation efficiency, especially for Gaussian kernels.

Findings

Achieves near-optimal query time and space complexity for radial kernels.

Uses geometric structure of sampled datasets to improve near neighbor search efficiency.

Provides data-dependent bounds that outperform previous worst-case analyses.

Abstract

In this paper we revisit the kernel density estimation problem: given a kernel $K(x, y)$ and a dataset of $n$ points in high dimensional Euclidean space, prepare a data structure that can quickly output, given a query $q$, a $(1+\epsilon)$-approximation to $\mu:=\frac1{|P|}\sum_{p\in P} K(p, q)$. First, we give a single data structure based on classical near neighbor search techniques that improves upon or essentially matches the query time and space complexity for all radial kernels considered in the literature so far. We then show how to improve both the query complexity and runtime by using recent advances in data-dependent near neighbor search. We achieve our results by giving a new implementation of the natural importance sampling scheme. Unlike previous approaches, our algorithm first samples the dataset uniformly (considering a geometric sequence of sampling rates), and then uses existing approximate near neighbor search techniques on the resulting smaller dataset to retrieve the sampled points that lie at an appropriate distance from the query. We show that the resulting sampled dataset has strong geometric structure, making approximate near neighbor search return the required samples much more efficiently than for worst case datasets of the same size. As an example application, we show that this approach yields a data structure that achieves query time $\mu^{-(1+o(1))/4}$ and space complexity $\mu^{-(1+o(1))}$ for the Gaussian kernel. Our data dependent approach achieves query time $\mu^{-0.173-o(1)}$ and space $\mu^{-(1+o(1))}$ for the Gaussian kernel. The data dependent analysis relies on new techniques for tracking the geometric structure of the input datasets in a recursive hashing process that we hope will be of interest in other applications in near neighbor search.