A Unified Matrix Factorization Framework for Classical and Robust Clustering
This work provides a principled framework for robust clustering, addressing sensitivity to outliers in data analysis, but it is incremental as it builds on existing matrix factorization interpretations.
The paper tackles the problem of unifying classical and robust clustering by reformulating crisp k-means and fuzzy c-means as matrix factorization problems, enabling robust variants using the l1,2-norm to handle outliers, with algorithms proven to converge locally.
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equivalence between crisp k-means clustering and matrix factorization, following and rigorously rederiving an unpublished formulation by Bauckhage.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.