  Recently, due to an increasing interest for transparency in artificial
intelligence, several methods of explainable machine learning have been
developed with the simultaneous goal of accuracy and interpretability by
humans. In this paper, we study a recent framework of explainable clustering
first suggested by Dasgupta et al.~\cite{dasgupta2020explainable}.
Specifically, we focus on the $k$-means and $k$-medians problems and provide
nearly tight upper and lower bounds.
  First, we provide an $O(\log k \log \log k)$-approximation algorithm for
explainable $k$-medians, improving on the best known algorithm of
$O(k)$~\cite{dasgupta2020explainable} and nearly matching the known
$\Omega(\log k)$ lower bound~\cite{dasgupta2020explainable}. In addition, in
low-dimensional spaces $d \ll \log k$, we show that our algorithm also provides
an $O(d \log^2 d)$-approximate solution for explainable $k$-medians. This
improves over the best known bound of $O(d \log k)$ for low
dimensions~\cite{laber2021explainable}, and is a constant for constant
dimensional spaces. To complement this, we show a nearly matching $\Omega(d)$
lower bound. Next, we study the $k$-means problem in this context and provide
an $O(k \log k)$-approximation algorithm for explainable $k$-means, improving
over the $O(k^2)$ bound of Dasgupta et al. and the $O(d k \log k)$ bound of
\cite{laber2021explainable}. To complement this we provide an almost tight
$\Omega(k)$ lower bound, improving over the $\Omega(\log k)$ lower bound of
Dasgupta et al. Given an approximate solution to the classic $k$-means and
$k$-medians, our algorithm for $k$-medians runs in time $O(kd \log^2 k )$ and
our algorithm for $k$-means runs in time $ O(k^2 d)$.
