基于密度约束近邻搜索的核密度估计
Kernel Density Estimation through Density Constrained Near Neighbor Search
摘要：
在本文中，我们重新审视核密度估计问题：给定一个核$K(x, y)$和一个高维欧氏空间中的$n$点的数据集，准备一个数据结构，可以快速输出，给定一个查询$q$，一个$(1+ε)$ -近似$μ:=\frac1{|P|}\sum_{p\in P} K(p, q)$。首先，我们给出了一个单一的数据结构的基础上，经典的近邻搜索技术，提高或基本上匹配的查询时间和空间复杂度的所有径向内核在文献中考虑到。然后，我们展示了如何提高查询的复杂性和运行时间，使用数据相关的近邻搜索的最新进展。我们实现了我们的结果，给出了一个新的实现自然重要性采样计划。与以前的方法不同，我们的算法首先统一采样数据集（考虑采样率的几何序列），然后使用现有的近似近邻搜索技术，得到的较小的数据集检索采样点，位于一个适当的距离查询。我们表明，由此产生的采样数据集具有强大的几何结构，使近似近邻搜索返回所需的样本比相同大小的最坏情况下的数据集更有效。作为一个例子的应用程序，我们表明，这种方法产生的数据结构，实现查询时间$μ^{-(1+o(1))/4}$和空间复杂度$μ^{-(1+o(1))}$的高斯内核。我们的数据依赖的方法实现了查询时间$μ^{-0.173-o(1)}$和空间$μ^{-(1+o(1))}$的高斯内核。数据相关分析依赖于新的技术，用于在递归散列过程中跟踪输入数据集的几何结构，我们希望这在近邻搜索的其他应用中会引起兴趣。
英文摘要：
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 $(1+ε)$-approximation to $μ:=\frac1{|P|}\sum_{p\in P} K(p, q)$. 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 $μ^{-(1+o(1))/4}$ and space complexity $μ^{-(1+o(1))}$ for the Gaussian kernel. Our data dependent approach achieves query time $μ^{-0.173-o(1)}$ and space $μ^{-(1+o(1))}$ 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.
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