The elbow statistic: Multiscale clustering statistical significance

• URL: http://arxiv.org/abs/2603.03235v1
• Date: Tue, 03 Mar 2026 18:28:01 GMT
• Title: The elbow statistic: Multiscale clustering statistical significance
• Authors: Francisco J. Perez-Reche,
• Abstract summary: We introduce ElbowSig, a framework that formalizes the elbow' method as a rigorous inferential problem.<n>As an algorithm-agnostic procedure, ElbowSig requires only the heterogeneity sequence and is compatible with a wide range of clustering methods.
• Score: 0.0
• License: http://creativecommons.org/licenses/by/4.0/
• Abstract: Selecting the number of clusters remains a fundamental challenge in unsupervised learning. Existing criteria typically target a single ``optimal'' partition, often overlooking statistically meaningful structure present at multiple resolutions. We introduce ElbowSig, a framework that formalizes the heuristic ``elbow'' method as a rigorous inferential problem. Our approach centers on a normalized discrete curvature statistic derived from the cluster heterogeneity sequence, which is evaluated against a null distribution of unstructured data. We derive the asymptotic properties of this null statistic in both large-sample and high-dimensional regimes, characterizing its baseline behavior and stochastic variability. As an algorithm-agnostic procedure, ElbowSig requires only the heterogeneity sequence and is compatible with a wide range of clustering methods, including hard, fuzzy, and model-based clustering. Extensive experiments on synthetic and empirical datasets demonstrate that the method maintains appropriate Type-I error control while providing the power to resolve multiscale organizational structures that are typically obscured by single-resolution selection criteria.

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