Clustering under Perturbation Resilience

TL;DR

This paper explores clustering algorithms under perturbation resilience, providing optimal solutions for certain objectives and developing new linkage criteria, with implications for real-world data where distances are heuristic.

Contribution

It introduces algorithms for perturbation-resilient clustering, including optimal solutions for center-based objectives and bounds for k-median under relaxed assumptions.

Findings

Optimal clustering for perturbation factor (1 + √2)

First bounds for k-median with small fraction change

Sublinear-time algorithms using random sampling

Abstract

Motivated by the fact that distances between data points in many real-world clustering instances are often based on heuristic measures, Bilu and Linial~\cite{BL} proposed analyzing objective based clustering problems under the assumption that the optimum clustering to the objective is preserved under small multiplicative perturbations to distances between points. The hope is that by exploiting the structure in such instances, one can overcome worst case hardness results. In this paper, we provide several results within this framework. For center-based objectives, we present an algorithm that can optimally cluster instances resilient to perturbations of factor $(1 + \sqrt{2})$, solving an open problem of Awasthi et al.~\cite{ABS10}. For $k$-median, a center-based objective of special interest, we additionally give algorithms for a more relaxed assumption in which we allow the optimal solution to change in a small $\epsilon$ fraction of the points after perturbation. We give the first bounds known for $k$-median under this more realistic and more general assumption. We also provide positive results for min-sum clustering which is typically a harder objective than center-based objectives from approximability standpoint. Our algorithms are based on new linkage criteria that may be of independent interest. Additionally, we give sublinear-time algorithms, showing algorithms that can return an implicit clustering from only access to a small random sample.