ICML 2025

Average Sensitivity of Hierarchical k-Median Clustering
Shijie Li 1 Weiqiang He 1 Ruobing Bai 1 Pan Peng 1
Abstract
Hierarchical clustering is a widely used method
for unsupervised learning with numerous applica-
tions. However, in the application of modern algo-
rithms, the datasets studied are usually large and
dynamic. If the hierarchical clustering is sensitive
to small perturbations of the dataset, the usabil-
ity of the algorithm will be greatly reduced. In
this paper, we focus on the hierarchical k -median
clustering problem, which bridges hierarchical
and centroid-based clustering while offering the-
oretical appeal, practical utility, and improved
interpretability. We analyze the average sensitiv-
ity of algorithms for this problem by measuring
the expected change in the output when a random
data point is deleted. We propose an efficient al-
gorithm for hierarchical k-median clustering and
theoretically prove its low average sensitivity and
high clustering quality. Additionally, we show
that single linkage clustering and a deterministic
variant of the CLNSS algorithm exhibit high aver-
age sensitivity, making them less stable. Finally,
we validate the robustness and effectiveness of
our algorithm through experiments.
1. Introduction
Hierarchical clustering, as discussed in (McQuitty, 1957;
Hastie et al., 2009), is one of the most commonly used
clustering strategies. It produces a set of nested clusters,
organized into a tree-like structure known as a dendrogram.
In this structure, each leaf node typically encompasses only
a single object, whereas each internal node represents the
union of its child nodes. By exploring the dendrogram’s
multiple levels, one can identify clusters with varying levels
of granularity. Consequently, hierarchical clustering is adept
at detecting and analyzing complex data structures. Hier-
archical clustering has attracted great interest in computer
1School of Computer Science and Technology, University of
Science and Technology of...