Stability of Partitions Induced by Nearest-Center Assignment Under Perturbations

TL;DR

This paper analyzes how partitions from nearest-center clustering change under data perturbations, identifying the margin as a key factor for stability and providing explicit conditions for invariance.

Contribution

It introduces a formal stability radius based on the margin and extends the analysis to dynamic and probabilistic settings, linking stability to the margin.

Findings

Partitions are stable if perturbations are less than half the minimum margin.

Instability occurs near decision boundaries where margin is small.

Small perturbations can still alter partitions, showing margin is sufficient but not necessary for stability.

Abstract

We study clustering through the partitions it induces on a finite labeled set $[n]=\{1,\dots,n\}$, and analyze how these partitions change under perturbations of a point configuration $X=(x_1,\dots,x_n)\in(\mathbb{R}^d)^n$. We equip the space of partitions $\Pi_n$ with a normalized pairwise disagreement metric $d(\cdot,\cdot)$, and define the stability radius $r(X,A)=\sup\{\varepsilon\ge0: A(X')=A(X)$ whenever $|X-X'\|\le\varepsilon\}$, where $\|X-X'\|=\max_i\|x_i-x'_i\|$. Our main results concern nearest-center assignment with fixed centers $\{c_1,\dots,c_k\}\subset\mathbb{R}^d$. For each point, we define the margin $\gamma_i=\min_{j\ne\ell_i}(\|x_i-c_j\|-\|x_i-c_{\ell_i}\|)$ and $\gamma_{\min}=\min_i\gamma_i$, where $\ell_i$ denotes the assigned center. We show that if $\varepsilon<\gamma_{\min}/2$, then no assignments change under perturbation and hence $A(X')=A(X)$. Conversely, any point that changes its assigned center must satisfy $\gamma_i\le2\varepsilon$, showing that instability is localized near decision boundaries. We construct configurations in which arbitrarily small perturbations $\|X-X'\|\le\varepsilon$ alter the induced partition, demonstrating that the margin condition is sufficient but not necessary for stability. We further extend the framework to a discrete-time setting, showing that if $\sum_t \delta_t<r(X(0),A)$, then $A(X(t))=A(X(0))$, and we give a probabilistic bound on $\mathbb{E}[d(A(X),A(X'))]$ in terms of tail probabilities relative to $\gamma_i$. This framework identifies the margin as the key quantity governing both worst-case and average stability, and provides explicit conditions under which clustering-induced partitions remain invariant in a fixed-center Euclidean model.