An upper bound on the silhouette evaluation metric for clustering
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866914410778329088 |
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| author | Sträng, Hugo Dinh, Tai |
| author_facet | Sträng, Hugo Dinh, Tai |
| contents | 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 for 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 for the macro-averaged silhouette. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2509_08625 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | An upper bound on the silhouette evaluation metric for clustering Sträng, Hugo Dinh, Tai Machine Learning 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 for 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 for the macro-averaged silhouette. |
| title | An upper bound on the silhouette evaluation metric for clustering |
| topic | Machine Learning |
| url | https://arxiv.org/abs/2509.08625 |