  $k$-means and $k$-median clustering are powerful unsupervised machine
learning techniques. However, due to complicated dependences on all the
features, it is challenging to interpret the resulting cluster assignments.
Moshkovitz, Dasgupta, Rashtchian, and Frost [ICML 2020] proposed an elegant
model of explainable $k$-means and $k$-median clustering. In this model, a
decision tree with $k$ leaves provides a straightforward characterization of
the data set into clusters.
  We study two natural algorithmic questions about explainable clustering. (1)
For a given clustering, how to find the "best explanation" by using a decision
tree with $k$ leaves? (2) For a given set of points, how to find a decision
tree with $k$ leaves minimizing the $k$-means/median objective of the resulting
explainable clustering? To address the first question, we introduce a new model
of explainable clustering. Our model, inspired by the notion of outliers in
robust statistics, is the following. We are seeking a small number of points
(outliers) whose removal makes the existing clustering well-explainable. For
addressing the second question, we initiate the study of the model of
Moshkovitz et al. from the perspective of multivariate complexity. Our rigorous
algorithmic analysis sheds some light on the influence of parameters like the
input size, dimension of the data, the number of outliers, the number of
clusters, and the approximation ratio, on the computational complexity of
explainable clustering.
