kM++kNN : A fast algorithm for the exact search of k-nearest neighbors

Raphael Lopes de Souza, Osvaldo Luiz De Oliveira
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Abstract

The k-NN algorithm - k-nearest neighbor - is widely used in Machine Learning and Statistics for tasks involving classification and regression. Having as inputs an instance x, a set of instances T and an integer $k \geqslant 1$, the k-NN performs an exhaustive search in T of the k instances most similar to instance x (k-nearest neighbors). In applications involving many instances and/or instances with high dimensionality, the search process is time-consuming due to the need to perform many calculations of similarity functions between instances. Several proposals to reduce the k-NN search time have been made, some of them aiming at the exact search of the k most similar instances to x in T and, others, reducing the search time via approximate methods to calculate the most similar instances to x. This work proposes an algorithm called $\mathrm{kM}++\mathrm{kNN}$ for the exact search of the k most similar instances to x in T, which uses the triangular inequality concept to reduce the ${\mathrm {k-N N}}$ search time. The ${\mathrm {k M++k N N}}$ algorithm is compared, in experiments to measure the economy of the number of calculations of similarity functions between instances and search time, with an algorithm currently considered fast, the kMkNN.
k++ kNN:精确搜索k个最近邻的快速算法
k-NN算法- k近邻-被广泛应用于机器学习和统计中涉及分类和回归的任务。以实例x、一组实例T和一个整数$k \geqslant 1$作为输入,k- nn在T中执行与实例x最相似的k个实例(k-近邻)的穷举搜索。在涉及许多实例和/或具有高维的实例的应用程序中,由于需要在实例之间执行许多相似性函数的计算,因此搜索过程非常耗时。已经提出了几个减少k- nn搜索时间的建议,其中一些建议旨在精确搜索T中与x最相似的k个实例,另一些建议通过近似方法计算与x最相似的实例来减少搜索时间。这项工作提出了一个名为$\mathrm{kM}++\mathrm{kNN}$的算法,用于精确搜索T中与x最相似的k个实例,该算法使用三角不等式概念来减少${\mathrm {k-N N}}$搜索时间。在实验中,将${\mathrm {k M++k N N}}$算法与目前被认为快速的kMkNN算法进行比较,以衡量实例之间相似函数的计算次数和搜索时间的经济性。
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