We consider the problem of explainable $k$-medians and $k$-means introduced by Dasgupta, Frost, Moshkovitz, and Rashtchian~(ICML 2020). In this problem, our goal is to find a \emph{threshold decision tree} that partitions data into $k$ clusters and minimizes the $k$-medians or $k$-means objective. The obtained clustering is easy to interpret because every decision node of a threshold tree splits data based on a single feature into two groups. We propose a new algorithm for this problem which is $\tilde O(\log k)$ competitive with $k$-medians with $\ell_1$ norm and $\tilde O(k)$ competitive with $k$-means. This is an improvement over the previous guarantees of $O(k)$ and $O(k^2)$ by Dasgupta et al (2020). We also provide a new algorithm which is $O(\log^{3/2} k)$ competitive for $k$-medians with $\ell_2$ norm. Our first algorithm is near-optimal: Dasgupta et al (2020) showed a lower bound of $\Omega(\log k)$ for $k$-medians; in this work, we prove a lower bound of $\tilde\Omega(k)$ for $k$-means. We also provide a lower bound of $\Omega(\log k)$ for $k$-medians with $\ell_2$ norm.

翻译：我们考虑的是Dasgupta、Frost、Moshkovitz和Rashtchian~(ICML 2020)引入的可解释的美元中值和美元中值问题。 在这个问题中,我们的目标是找到一个将数据分割成美元组的 emph{threswork 决策树}, 将美元中值或美元中值或美元中值的目标最小化。 获得的分组很容易解释, 因为一个树阈值的每个决定节点将基于一个特性的数据分割成两个组。 我们为此问题提出了一个新的算法, 即$\tilde O(log k) $和美元中值中$1美元中正值中值中, 以美元中值中值为美元中值中值中值中值中, 美元中以美元中值中值中值中, 美元中以美元中值中, 美元中以美元中正值中值中, 美元中以美元中正值中为美元中。