An upper bound on the silhouette evaluation metric for clustering

• Derives a global, data-dependent upper bound for average silhouette width (ASW). • Gives pointwise silhouette-width bounds, and aggregates them to the ASW ceiling. • Computes bounds from pairwise dissimilarities in O(n2 log n) time overall only. • Experiments show bounds aid ASW interpretation; usefulness is datasetdependent. • Extends the framework to an upper bound for the macro-averaged silhouette score. The silhouette coefficient quantifies, for each observation, the balance between within-cluster cohesion and between-cluster separation, taking values in the range [ − 1 , 1 ] . The average silhouette width ( ASW ) is a widely used internal measure of clustering quality, with higher values indicating more cohesive and well-separated clusters. However, the dataset-specific maximum of ASW is typically unknown, and the standard upper limit of 1 is rarely attainable. In this work, we derive for each data point a sharp upper bound on its silhouette width and aggregate these to obtain a canonical upper bound of the ASW . This bound—often substantially below 1—enhances the interpretability of empirical ASW values by providing guidance on how close a given clustering result is to the best possible outcome for that dataset. We evaluate the usefulness of the upper bound on a variety of datasets and conclude that it can meaningfully enrich cluster quality evaluation; however, its practical relevance depends on the specific dataset. Finally, we extend the framework to establish an upper bound of the macro-averaged silhouette.