A Unified Matrix Factorization Framework for Classical and Robust Clustering

TL;DR

This paper unifies classical and robust clustering under a matrix factorization framework, extending it to fuzzy clustering and introducing robust variants that handle outliers effectively.

Contribution

It formalizes fuzzy c-means as a matrix factorization problem and develops robust formulations with convergence guarantees.

Findings

Unified framework for classical and robust clustering

New matrix factorization interpretation for fuzzy c-means

Algorithms with proven convergence to local minima

Abstract

This paper presents a unified matrix factorization framework for classical and robust clustering. We begin by revisiting the well-known equivalence between crisp k-means clustering and matrix factorization, following and rigorously rederiving an unpublished formulation by Bauckhage. Extending this framework, we derive an analogous matrix factorization interpretation for fuzzy c-means clustering, which to the best of our knowledge has not been previously formalized. These reformulations allow both clustering paradigms to be expressed as optimization problems over factor matrices, thereby enabling principled extensions to robust variants. To address sensitivity to outliers, we propose robust formulations for both crisp and fuzzy clustering by replacing the Frobenius norm with the l1,2-norm, which penalizes the sum of Euclidean norms across residual columns. We develop alternating minimization algorithms for the standard formulations and IRLS-based algorithms for the robust counterparts. All algorithms are theoretically proven to converge to a local minimum.