Impossibility of Depth Reduction in Explainable Clustering
Over the last few years Explainable Clustering has gathered a lot of attention. Dasgupta et al. [ICML'20] initiated the study of explainable k-means and k-median clustering problems where the explanation is captured by a threshold decision tree which partitions the space at each node using axis...
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Zusammenfassung: | Over the last few years Explainable Clustering has gathered a lot of
attention. Dasgupta et al. [ICML'20] initiated the study of explainable k-means
and k-median clustering problems where the explanation is captured by a
threshold decision tree which partitions the space at each node using axis
parallel hyperplanes. Recently, Laber et al. [Pattern Recognition'23] made a
case to consider the depth of the decision tree as an additional complexity
measure of interest.
In this work, we prove that even when the input points are in the Euclidean
plane, then any depth reduction in the explanation incurs unbounded loss in the
k-means and k-median cost. Formally, we show that there exists a data set X in
the Euclidean plane, for which there is a decision tree of depth k-1 whose
k-means/k-median cost matches the optimal clustering cost of X, but every
decision tree of depth less than k-1 has unbounded cost w.r.t. the optimal cost
of clustering. We extend our results to the k-center objective as well, albeit
with weaker guarantees. |
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DOI: | 10.48550/arxiv.2305.02850 |