  We study the problem of explainable clustering in the setting first
formalized by Dasgupta, Frost, Moshkovitz, and Rashtchian (ICML 2020). A
$k$-clustering is said to be explainable if it is given by a decision tree
where each internal node splits data points with a threshold cut in a single
dimension (feature), and each of the $k$ leaves corresponds to a cluster. We
give an algorithm that outputs an explainable clustering that loses at most a
factor of $O(\log^2 k)$ compared to an optimal (not necessarily explainable)
clustering for the $k$-medians objective, and a factor of $O(k \log^2 k)$ for
the $k$-means objective. This improves over the previous best upper bounds of
$O(k)$ and $O(k^2)$, respectively, and nearly matches the previous $\Omega(\log
k)$ lower bound for $k$-medians and our new $\Omega(k)$ lower bound for
$k$-means. The algorithm is remarkably simple. In particular, given an initial
not necessarily explainable clustering in $\mathbb{R}^d$, it is oblivious to
the data points and runs in time $O(dk \log^2 k)$, independent of the number of
data points $n$. Our upper and lower bounds also generalize to objectives given
by higher $\ell_p$-norms.
