Composite Silhouette: A Subsampling-based Aggregation Strategy

Abstract: Determining the number of clusters is a central challenge in unsupervised learning, where ground-truth labels are unavailable. The Silhouette coefficient is a widely used internal validation metric for this task, yet its standard micro-averaged form tends to favor larger clusters under size imbalance. Macro-averaging mitigates this bias by weighting clusters equally, but may overemphasize noise from under-represented groups. We introduce Composite Silhouette, an internal criterion for cluster-count selection that aggregates evidence across repeated subsampled clusterings rather than relying on a single partition. For each subsample, micro- and macro-averaged Silhouette scores are combined through an adaptive convex weight determined by their normalized discrepancy and smoothed by a bounded nonlinearity; the final score is then obtained by averaging these subsample-level composites. We establish key properties of the criterion and derive finite-sample concentration guarantees for its subsampling estimate. Experiments on synthetic and real-world datasets show that Composite Silhouette effectively reconciles the strengths of micro- and macro-averaging, yielding more accurate recovery of the ground-truth number of clusters.

• The paper introduces Composite Silhouette, a novel framework for adaptive cluster-count selection using discrepancy-aware subsampling and aggregation.
• It leverages weighted micro- and macro-averaged Silhouette scores to mitigate biases from imbalanced and heterogeneous data conditions.
• Empirical and theoretical results confirm rapid convergence and reliable recovery of the ground-truth number of clusters across diverse datasets.

Composite Silhouette: Subsampling-Based Adaptive Aggregation for Internal Cluster Validation

Motivation and Problem Statement

Cluster-count selection remains a prominent challenge in unsupervised learning, particularly in domains with substantial cluster-size imbalance or structural heterogeneity where conventional internal validation indices often yield inconsistent or biased model selection. The Silhouette coefficient, a widely adopted metric, is typically aggregated via micro-averaging—favoring larger clusters—or macro-averaging—emphasizing smaller clusters. Neither approach consistently yields optimal results across diverse data regimes, leading to fundamental inductive bias conflicts Figure 1.

Figure 1: Illustration of the trade-off between global micro-averaging and local macro-averaging for cluster validation, highlighting domain impact and bias conflict.

The key issue stems from the absence of principled strategies for balancing these aggregation perspectives in cluster-count selection tasks where structural conditions vary across datasets and applications. Micro-averaging is susceptible to overweighting major groups, while macro-averaging may amplify noise from minor clusters. This motivates a data-adaptive criterion capable of leveraging both perspectives without global preference.

Methodological Framework

Composite Silhouette ($S_\mathrm{mM}$) is formulated as an internal criterion employing repeated subsampling and discrepancy-aware aggregation. For each candidate cluster count $k$, the method performs $B$ independent subsampled clusterings, computing micro- ($S_\mathrm{m}^{(b)}$) and macro- ($S_\mathrm{M}^{(b)}$) averaged Silhouette scores per subsample. The normalized discrepancy $\widetilde{\Delta}_b$ between these scores is transformed via a bounded nonlinearity ($\tanh$), yielding a convex weight $w_b$ for adaptive combination. The overall composite score for $k$ is then the average over $B$ such weighted subsample-level scores:

$k$0

where $k$1.

This convex combination ensures responsiveness to local structural variation: subsamples where micro- and macro-views diverge are adaptively weighted based on their observed discrepancy, and the final composite succinctly encapsulates their aggregate behavior across the data.

Figure 2: Approximation error of $k$2 as a function of the number of subsamples $k$3.

Theoretical Properties and Guarantees

The paper establishes deterministic properties for $k$4, including that each subsample-level score lies between the corresponding micro- and macro-averaged Silhouette scores, and that the composite differs from globally averaged scores by a discrepancy-dependent correction. Hoeffding-type concentration bounds are derived for the subsampling estimate, showing that $k$5 converges to its expectation as $k$6 with $k$7 rate, both for fixed $k$8 and uniformly over the candidate set. Under a positive-margin condition on the expected composite scores, the probability of recovering the optimal $k$9 approaches unity as subsample count increases.

Empirical Evaluation: Synthetic and Real-World Benchmarks

Composite Silhouette's efficacy is validated across four synthetic datasets (S1–S4) representing a progression from balanced, well-separated clusters to complex, heterogeneous, and strongly imbalanced regimes; as well as twelve real-world datasets spanning tabular, image, and text domains, with varied cluster counts and structures.

The results demonstrate that $B$0 consistently recovers the ground-truth number of clusters, outperforming both micro- and macro-averaged baselines, Calinski–Harabasz, Davies–Bouldin, Elbow, and Gap statistics. In strongly imbalanced scenarios (S3, B77), $B$1 leverages macro-averaging when it aligns with ground truth; in diffuse or ambiguous cases (S4, Nsg, Mds), it adaptively balances both views to achieve robust selection.

Figure 3: Visualization of S1 synthetic data benchmark, colored by ground-truth clusters.

Figure 4: Visualization of S4 synthetic data with structural heterogeneity; cluster size and scale vary dramatically.

Composite Silhouette not only matches or surpasses baseline selection accuracy (Table: paper Table 1), but maintains strong values at ground-truth cluster counts across all dataset types. Its approximation error decays rapidly with modest $B$2 (10–30 sufficient), and subsample size heuristics allow for scalable application without per-dataset tuning.

Figure 5: Runtime scaling of $B$3 versus standard full-data Silhouette score as a function of sample size $B$4.

Implementation and Practical Considerations

The method’s complexity is dominated by $B$5 subsampled clusterings and Silhouette computations, with negligible overhead from discrepancy-aware weighting. The proposed automatic subsample-size selection rule ensures robust cluster representation and computational tractability. Composite Silhouette is compatible with parallelization, facilitating efficient operation on large datasets.

The $B$6 transformation for discrepancy is empirically validated as the most reliable, providing stable cluster-count recovery where linear or step mappings fail. The code is released openly for community adoption.

Implications and Future Directions

Composite Silhouette furnishes a principled, adaptive approach to internal cluster validation, with immediate applicability to model selection in unsupervised learning. Its discrepancy-aware aggregation is robust to structural imbalance and heterogeneity, obviating the need for domain-specific aggregation preference.

Theoretically, this paradigm invites extensions to other validation indices and clustering paradigms, including hierarchical and density-based methods. Potential future directions include systematic characterization of regimes where composite aggregation outperforms both constituent views, investigation into stability for non-Euclidean dissimilarities, and exploration of meta-learning mechanisms for optimizing subsample and aggregation protocols.

Conclusion

Composite Silhouette ($B$7) represents a significant advance in internal cluster validation, delivering consistent and robust cluster-count selection across diverse real and synthetic domains through subsampling and discrepancy-adaptive aggregation. This framework enables practitioners to reconcile the strengths of micro- and macro-averaging without prior assumptions, offering a practical, scalable, and theoretically grounded strategy for unsupervised model selection (2604.13816).