  This paper explores hierarchical clustering in the case where pairs of points
have dissimilarity scores (e.g. distances) as a part of the input. The recently
introduced objective for points with dissimilarity scores results in every tree
being a 1/2 approximation if the distances form a metric. This shows the
objective does not make a significant distinction between a good and poor
hierarchical clustering in metric spaces. Motivated by this, the paper develops
a new global objective for hierarchical clustering in Euclidean space. The
objective captures the criterion that has motivated the use of divisive
clustering algorithms: that when a split happens, points in the same cluster
should be more similar than points in different clusters. Moreover, this
objective gives reasonable results on ground-truth inputs for hierarchical
clustering. The paper builds a theoretical connection between this objective
and the bisecting k-means algorithm. This paper proves that the optimal 2-means
solution results in a constant approximation for the objective. This is the
first paper to show the bisecting k-means algorithm optimizes a natural global
objective over the entire tree.
