论文标题
$ k $ -nn规则在公制空间和长田维度的普遍一致性
Universal consistency of the $k$-NN rule in metric spaces and Nagata dimension
论文作者
论文摘要
$ k $最近的邻居学习规则（在统一距离领带的破裂下）在每一个度量$ x $中都是一致的。 Cérou和Guyader（2006）指出了这一点，这是这些作者的主要结果，并结合了D. Preiss（1971）绘制的真实分析中的定理（并由Assouad和Quentin de Gromard（2006）详细详细介绍）。我们表明，可以与Charles J. Stone（1977）的原始定理进行直接证明，介绍$ K $ -NN分类器在有限的维度欧几里得空间中的普遍一致性。概括是非平凡的，因为距离在非欧国人的设置中更为普遍，并且在我们研究指标的相关几何特性和石头参数的局限性上，通过构造各种示例。
The $k$ nearest neighbour learning rule (under the uniform distance tie breaking) is universally consistent in every metric space $X$ that is sigma-finite dimensional in the sense of Nagata. This was pointed out by Cérou and Guyader (2006) as a consequence of the main result by those authors, combined with a theorem in real analysis sketched by D. Preiss (1971) (and elaborated in detail by Assouad and Quentin de Gromard (2006)). We show that it is possible to give a direct proof along the same lines as the original theorem of Charles J. Stone (1977) about the universal consistency of the $k$-NN classifier in the finite dimensional Euclidean space. The generalization is non-trivial because of the distance ties being more prevalent in the non-euclidean setting, and on the way we investigate the relevant geometric properties of the metrics and the limitations of the Stone argument, by constructing various examples.
