下兔子洞：次线性空间中的鲁棒邻近搜索和密度估计
Down the Rabbit Hole: Robust Proximity Search and Density Estimation in Sublinear Space
摘要：
对于$\Re^d$中的一组$n$点，以及参数$k$和$\eps$，我们提出了一种数据结构，可以在对数时间内回答$(1+\eps,k)$ - \ANN查询。令人惊讶的是，数据结构使用的空间是$\Otilde (n /k)$ ;也就是说，如果$k$足够大，则所使用的空间在输入大小上是次线性的。我们的方法提供了一种新的方式来总结几何数据，这样有意义的接近查询的数据可以使用这个草图。使用这一点，我们提供了一个次线性空间数据结构，可以估计密度的点集下的各种措施，包括：\begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum}我们的方法推广到其他距离为基础的估计密度相似的味道。我们还研究了使用采样时近似这些数量的问题。特别是，我们表明，一个样本的大小$\Otilde (n /k)$是足够的，在某些限制的情况下，估计上述数量。值得注意的是，样本量与维度只有线性关系。
英文摘要：
For a set of $n$ points in $\Re^d$, and parameters $k$ and $\eps$, we present a data structure that answers $(1+\eps,k)$-\ANN queries in logarithmic time. Surprisingly, the space used by the data-structure is $\Otilde (n /k)$; that is, the space used is sublinear in the input size if $k$ is sufficiently large. Our approach provides a novel way to summarize geometric data, such that meaningful proximity queries on the data can be carried out using this sketch. Using this, we provide a sublinear space data-structure that can estimate the density of a point set under various measures, including: \begin{inparaenum}[(i)] \item sum of distances of $k$ closest points to the query point, and \item sum of squared distances of $k$ closest points to the query point. \end{inparaenum} Our approach generalizes to other distance based estimation of densities of similar flavor. We also study the problem of approximating some of these quantities when using sampling. In particular, we show that a sample of size $\Otilde (n /k)$ is sufficient, in some restricted cases, to estimate the above quantities. Remarkably, the sample size has only linear dependency on the dimension.
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