Clustering results should express uncertainty about cluster numbers
Bracketing Uncertainty in Clustering Under the Manifold Hypothesis
Machine Learning
Summary
Clustering is about grouping data points that are similar, but sometimes it’s unclear how many groups there really are. The authors studied this problem by looking at the geometry of data that lies on curved shapes called manifolds. They found that in many cases there isn’t enough information to pick one clear number of clusters. Their new method doesn’t force a single answer but instead gives a range showing how uncertain the number of groups is, helping users know when more data or caution is needed.
What this means in practice
- •For data science teams: Assess clustering outcomes with uncertainty brackets to avoid overconfident conclusions about the number of groups in complex datasets.
- •For machine learning engineers: Incorporate manifold-aware clustering diagnostics to improve model tuning when data geometry suggests ambiguous cluster counts.
Authors
Savik Kinger, Luciano Dyballa, Steven W. Zucker
Abstract
The manifold hypothesis suggests a natural criterion for clustering: partition data according to the manifold component from which each point is drawn. Whether two components are separable depends on a geometric tradeoff: the ambient separation between components versus the largest gap in sampling. In practice, this tradeoff is rarely assessed explicitly, leading standard methods to over-commit to a single clustering assignment even when the data do not support a unique answer. We formalize this tradeoff by combining intrinsic manifold geometry (volume growth and reach) with sample-level quantities (fill distance and density), yielding a threshold phenomenon for mutual-$k$-nearest-neighbor graphs: when the offset-to-fill ratio exceeds a conservative upper threshold, component separation is preserved; below a lower threshold, components fuse. The gap between these thresholds defines a geometric uncertainty zone in which the number of clusters is not identifiable from the data. Nevertheless, conventional approaches still seek one: sweeping parameters (an engineering approach) or fitting a generative mixture model (a model-based approach). Rather than forcing a single estimate of the number of clusters, we propose Manifold-Based Clustering (MBC), which returns an explicit bracket interval to quantify the underlying data uncertainty. This bracket acts as an empirically calibrated diagnostic: it narrows when a single resolution is supported, widens when multiple resolutions coexist, and collapses to one when no separated structure is detectable. Empirically, we find that many real datasets lie within the uncertainty zone rather than admitting one clear answer. Our results suggest that ambiguity in cluster number is often intrinsic, and should be quantified rather than resolved.