An Objective for Hierarchical Clustering in Euclidean Space and its Connection to Bisecting K-means

• URL: http://arxiv.org/abs/2008.13235v1
• Date: Sun, 30 Aug 2020 18:17:46 GMT
• Title: An Objective for Hierarchical Clustering in Euclidean Space and its Connection to Bisecting K-means
• Authors: Benjamin Moseley, Yuyan Wang
• Abstract summary: Recently introduced objective for points with dissimilarity scores results in every tree being a 1/2 approximation if the distances form a metric. Motivated by this, the paper develops a new global objective for hierarchical clustering in Euclidean space. The paper builds a theoretical connection between this objective and the bisecting k-means algorithm.
• Score: 20.322355919030112
• License: http://arxiv.org/licenses/nonexclusive-distrib/1.0/
• Abstract: 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.

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