A sharp local threshold is identified, stability under bounded perturbations is proved, monotonicity in $k$ is established, and an intrinsic $\Theta(\log n)$ separation between local and global realizability is shown.
Abstract
Hierarchical clustering produces ultrametric representations that impose strong global geometric constraints and may distort local similarities in ways that disproportionately affect individual data points. We study hierarchical clustering under an individual fairness requirement that bounds relative distortion within local $k$-nearest neighborhoods. We formulate this requirement as a feasibility problem over dominated ultrametrics and characterize the minimal multiplicative slack required for feasibility. We identify a sharp local threshold, prove stability under bounded perturbations, establish monotonicity in $k$, and show an intrinsic $\Theta(\log n)$ separation between local and global realizability. Experiments on synthetic and real world datasets support our theoretical results.
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