Assigning Confidence: K-partition Ensembles

TL;DR

CAKE (Confidence in Assignments via K-partition Ensembles), a framework that evaluates each point using two complementary statistics computed over a clustering ensemble: assignment stability and consistency of local geometric fit, is introduced.

Abstract

Clustering is widely used for unsupervised structure discovery, yet it offers limited insight into how reliable each individual assignment is. Diagnostics, such as convergence behavior or objective values, may reflect global quality, but they do not indicate whether particular instances are assigned confidently, especially for initialization-sensitive algorithms like k-means. This assignment-level instability can undermine both accuracy and robustness. Ensemble approaches improve global consistency by aggregating multiple runs, but they typically lack tools for quantifying pointwise confidence in a way that combines cross-run agreement with geometric support from the learned cluster structure. We introduce CAKE (Confidence in Assignments via K-partition Ensembles), a framework that evaluates each point using two complementary statistics computed over a clustering ensemble: assignment stability and consistency of local geometric fit. These are combined into a single, interpretable score in [0,1]. Our theoretical analysis shows that CAKE remains effective under noise and separates stable from unstable points. Experiments on synthetic and real-world datasets indicate that CAKE effectively highlights ambiguous points and stable core members, providing a confidence ranking that can guide filtering or prioritization to improve clustering quality.

Figures (13)

• Figure 1: Assignment stability and geometric consistency can fail in complementary ways. P1 (stable outlier; left) is consistently assigned (high stability) despite weak integration into its assigned cluster structure, while P2 (unstable boundary; right) shows higher within-cluster fit to one cluster in a single run yet switches labels across runs near the boundary. These cases suggest that reliable pointwise confidence should account for both signals jointly.
• Figure 2: CAKE framework overview. Across an ensemble of $R$ clustering runs (partitions), CAKE aggregates Silhouette statistics into a geometric component and (aligned) label assignments into a stability component, then fuses the two into a confidence score for each data point.
• Figure 3: $\mathrm{CAKE}^{\text{(PR)}}$ (Eq. \ref{['eq:cake:a']}) scores distribution (left) and percentiles (right) on synthetic data.
• Figure 4: $\mathrm{CAKE}^{\text{(PR)}}$ on the two-moons dataset ($n{=}1200$, noise $=0.06$) with $3\%$ uniform outliers. We construct an ensemble of $R{=}25$ partitions using spectral clustering on a $k$-NN graph, jittering $n_{\text{neighbors}}\in[10,15]$ per run to induce diversity. For the geometric component we use the kernelized Silhouette (§\ref{['subsec:silhouette']}; Eq. \ref{['eq:kerneldist']}) with a self-tuning RBF kernel: $\kappa(x_i,x_j)=\exp\!(-\|x_i-x_j\|^2/(\sigma_i\sigma_j))$. We build the Gram matrix $K$ once and reuse it across runs, setting each $\sigma_i$ to the distance from $x_i$ to its $k_{\text{nn}}{=}7$th nearest neighbor. Points are colored by their CAKE scores ($\uparrow$red, $\downarrow$blue).
• Figure 5: Synthetic datasets (S1-S7), colored by ground-truth cluster labels; noise in light gray.