# An upper bound on the silhouette evaluation metric for clustering

arXiv: https://arxiv.org/abs/2509.08625 · HTML: https://mlsanity.com/paper/2509.08625/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 11 of its 111 references that are in the map.
- First posted on arXiv 10 Sep 2025.

## Builds on (most cited first, 11 of 11)

- [Faster k-Medoids Clustering: Improving the PAM, CLARA, and CLARANS Algorithms](https://mlsanity.com/paper/1810.05691/) (2018)
- [Are Cluster Validity Measures (In)valid?](https://mlsanity.com/paper/2208.01261/) (2022)
- [Clustering performance analysis using a new correlation-based cluster validity index](https://mlsanity.com/paper/2109.11172/) (2021)
- [Estimating the Optimal Number of Clusters in Categorical Data Clustering by Silhouette Coefficient](https://mlsanity.com/paper/2501.15542/) (2025)
- [A Self-Supervised Learning-based Approach to Clustering Multivariate Time-Series Data with Missing Values (SLAC-Time): An Application to TBI Phenotyping](https://mlsanity.com/paper/2302.13457/) (2023)
- [Medoid Silhouette clustering with automatic cluster number selection](https://mlsanity.com/paper/2309.03751/) (2023)
- [Revisiting Silhouette Aggregation](https://mlsanity.com/paper/2401.05831/) (2024)
- [Clustering by Direct Optimization of the Medoid Silhouette](https://mlsanity.com/paper/2209.12553/) (2022)
- [Improving clustering quality evaluation in noisy Gaussian mixtures](https://mlsanity.com/paper/2503.00379/) (2025)
- [Data clustering: a fundamental method in data science and management](https://mlsanity.com/paper/2412.18760/) (2024)
- [On estimation and order selection for multivariate extremes via clustering](https://mlsanity.com/paper/2406.14535/) (2024)

## Abstract (arXiv)

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.