用多重排序法有效构造邻域图
Efficient Construction of Neighborhood Graphs by the Multiple Sorting Method
摘要：
邻域图作为机器学习中一种简洁的数据表示方法越来越受欢迎。然而，通过成对距离计算的朴素图构造对于$n$数据点花费$O(n^2)$运行时间，并且这对于数百万个数据点是非常慢的。对于等长的字符串，多重排序方法（Uno，2008）可以在$O(n+m)$时间内构建$ε$ -邻居图，其中$m$是数据中$ε$ -邻居对的数量。为了将这种非常有效的算法引入到图像、信号和文本等连续域，我们采用了随机投影方法将向量转换为字符串。理论结果阐明近似质量和计算时间之间的权衡。实证结果表明，我们的方法相比，快速最近邻替代品的效率。
英文摘要：
Neighborhood graphs are gaining popularity as a concise data representation in machine learning. However, naive graph construction by pairwise distance calculation takes $O(n^2)$ runtime for $n$ data points and this is prohibitively slow for millions of data points. For strings of equal length, the multiple sorting method (Uno, 2008) can construct an $ε$-neighbor graph in $O(n+m)$ time, where $m$ is the number of $ε$-neighbor pairs in the data. To introduce this remarkably efficient algorithm to continuous domains such as images, signals and texts, we employ a random projection method to convert vectors to strings. Theoretical results are presented to elucidate the trade-off between approximation quality and computation time. Empirical results show the efficiency of our method in comparison to fast nearest neighbor alternatives.
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