An upper bound of the silhouette validation metric for clustering

Abstract: The silhouette coefficient summarizes, per observation, cohesion versus separation in [-1, 1]; the average silhouette width (ASW) is a common internal measure of clustering quality where higher values indicate more coveted results. However, the dataset-specific maximum of ASW is typically unknown, and the standard upper limit 1 is often unattainable. In this work, we derive for each data point in a given dataset a sharp upper bound on its silhouette width. By aggregating these individual bounds, we present a canonical data-dependent upper bound on ASW that often assumes values well below 1. The presented bounds can indicate whether individual data points can ever be well placed, enable early stopping of silhouette-based optimization loops, and help answer a key question: How close is my clustering result to the best possible outcome on this specific data? Across synthetic and real datasets, the bounds are provably near-tight in many cases and offer significant enrichment of cluster quality evaluation.