  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 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.
