In nearest-neighbor classification, a training set $P$ of points in $\mathbb{R}^d$ with given classification is used to classify every point in $\mathbb{R}^d$: Every point gets the same classification as its nearest neighbor in $P$. Recently, Eppstein [SOSA'22] developed an algorithm to detect the relevant training points, those points $p\in P$, such that $P$ and $P\setminus\{p\}$ induce different classifications. We investigate the problem of finding the minimum cardinality reduced training set $P'\subseteq P$ such that $P$ and $P'$ induce the same classification. We show that the set of relevant points is such a minimum cardinality reduced training set if $P$ is in general position. Furthermore, we show that finding a minimum cardinality reduced training set for possibly degenerate $P$ is in P for $d=1$, and NP-complete for $d\geq 2$.

翻译：在最近邻分类中，一个带有给定分类的 $\mathbb{R}^d$ 中点集 $P$ 被用作训练集，用于对 $\mathbb{R}^d$ 中的每个点进行分类：每个点的分类与其在 $P$ 中的最近邻相同。最近，Eppstein [SOSA'22] 提出了一种算法，用于检测相关训练点，即那些使得 $P$ 和 $P\setminus\{p\}$ 引发不同分类的点 $p\in P$。我们研究了寻找最小基数缩减训练集 $P'\subseteq P$ 的问题，使得 $P$ 和 $P'$ 引发相同的分类。我们证明，如果 $P$ 处于一般位置，则相关点集就是一个这样的最小基数缩减训练集。此外，我们证明，对于可能退化的 $P$，当 $d=1$ 时，寻找最小基数缩减训练集属于 P 类问题，而当 $d\geq 2$ 时，该问题为 NP-完全问题。