Revisiting Silhouette Aggregation

TL;DR

The paper addresses the vulnerability of micro-averaged Silhouette scores to cluster imbalance and argues that macro-averaging—though underused—offers greater robustness. It formalizes both aggregation strategies, introduces per-cluster sampling to enable robust macro-aggregation, and empirically validates the approach on synthetic and eight real-world datasets. The key contributions are a formal comparison of MiS and MaS, the per-cluster sampling method, and evidence that macro-averaging improves k-estimation and clustering evaluation in imbalanced settings. The work provides practical guidance for practitioners and includes publicly available code, advancing reliable internal validation in clustering tasks for domains where imbalance is common.

Abstract

Silhouette coefficient is an established internal clustering evaluation measure that produces a score per data point, assessing the quality of its clustering assignment. To assess the quality of the clustering of the whole dataset, the scores of all the points in the dataset are typically (micro) averaged into a single value. An alternative path, however, that is rarely employed, is to average first at the cluster level and then (macro) average across clusters. As we illustrate in this work with a synthetic example, the typical micro-averaging strategy is sensitive to cluster imbalance while the overlooked macro-averaging strategy is far more robust. By investigating macro-Silhouette further, we find that uniform sub-sampling, the only available strategy in existing libraries, harms the measure's robustness against imbalance. We address this issue by proposing a per-cluster sampling method. An experimental study on eight real-world datasets is then used to analyse both coefficients in two clustering tasks.

Figures (8)

• Figure 1: Illustration of the elements involved in the computation of the silhouette score $s(x_i)$ for a given data point $x_i$ that belongs to cluster $C_I$.
• Figure 2: Synthetic dataset, shown on the left, with four equibalanced clusters. The same space is shown on the right, but the relatively distant cluster B now comprises 5,000 points more, yielding a heavy cluster-imbalance. Silhouette is reported per dataset per aggregation strategy. Micro-averaging increases unreasonably by a large margin.
• Figure 3: Silhouette score of the dataset of Figure \ref{['fig:synthetic']}, micro- and macro-averaged, for a varying number of points added.
• Figure 4: Macro-averaged Silhouette score, computed on uniform and per-cluster samples of 100 points, as the size of cluster B of Figure \ref{['fig:synthetic']} increases.
• Figure 5: Synthetic dataset, equibalanced as in Figure \ref{['fig:synthetic']} on the left, but now cluster B (to which we add 5,000 points on the right) is very close to clusters D and A.