Optimal Clustering under Uncertainty

TL;DR

This paper develops a probabilistic framework for optimal robust clustering under uncertainty, extending classical methods to incorporate randomness and improve performance in applications like granular imaging.

Contribution

It introduces a new robust clustering approach based on effective random point processes that handle uncertainty and derives an optimal clusterer within this framework.

Findings

Robust clusterers outperform traditional methods in synthetic Gaussian mixtures.

The framework successfully applies to granular imaging using granulometric moment theory.

Performance evaluations demonstrate improved accuracy under uncertainty.

Abstract

Classical clustering algorithms typically either lack an underlying probability framework to make them predictive or focus on parameter estimation rather than defining and minimizing a notion of error. Recent work addresses these issues by developing a probabilistic framework based on the theory of random labeled point processes and characterizing a Bayes clusterer that minimizes the number of misclustered points. The Bayes clusterer is analogous to the Bayes classifier. Whereas determining a Bayes classifier requires full knowledge of the feature-label distribution, deriving a Bayes clusterer requires full knowledge of the point process. When uncertain of the point process, one would like to find a robust clusterer that is optimal over the uncertainty, just as one may find optimal robust classifiers with uncertain feature-label distributions. Herein, we derive an optimal robust clusterer by first finding an effective random point process that incorporates all randomness within its own probabilistic structure and from which a Bayes clusterer can be derived that provides an optimal robust clusterer relative to the uncertainty. This is analogous to the use of effective class-conditional distributions in robust classification. After evaluating the performance of robust clusterers in synthetic mixtures of Gaussians models, we apply the framework to granular imaging, where we make use of the asymptotic granulometric moment theory for granular images to relate robust clustering theory to the application.