为聚类评估指标提供可解释的理论上限，帮助判断结果离最优有多远。
An upper bound on the silhouette evaluation metric for clustering
- 为每个数据点计算精确的轮廓系数上限，聚合得到整体上界。
- 实际数据集的上限常远低于1，显著低于传统认知。
- 适用于想精准评估聚类质量的研究者，尤其在对比不同算法时。
轮廓系数衡量每个样本在簇内凝聚度与簇间分离度之间的平衡，取值范围为[-1,1]。平均轮廓宽度（ASW）是广泛使用的聚类质量内部评估指标，值越高表示聚类越紧密且分离越好。然而，数据集特有的ASW最大值通常未知，标准上限1极少可达。本文针对每个数据点推导出其轮廓系数的严格上界，并聚合得到ASW的规范上界。该上界——通常显著低于1——通过提供与数据集最优结果的距离指引，增强了实测ASW值的可解释性。我们在多种数据集上验证了该上界的实用性，结论表明它能有效提升聚类质量评估的深度；但其实际意义依赖于具体数据集特征。最后，我们将框架扩展至宏平均轮廓系数的上界推导。
原文摘要 · Abstract (English)
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.
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