统一矩阵分解框架，让传统与抗噪聚类无缝衔接。
A Unified Matrix Factorization Framework for Classical and Robust Clustering
- 用矩阵分解重写硬聚类和模糊聚类，统一建模思路。
- 引入l1,2范数降低异常值影响，显著提升鲁棒性。
- 算法收敛有保证，适合数据含噪声的场景使用。
本文提出一个统一的矩阵分解框架，涵盖经典聚类与鲁棒聚类。首先重新审视硬k-means与矩阵分解之间的等价关系，并严谨推导出由Bauckhage未发表的表述所启发的公式。进一步，首次形式化地建立了模糊c-means聚类的矩阵分解解释。这些重构使两类聚类均能表示为因子矩阵上的优化问题，从而可自然推广至鲁棒版本。为应对异常值敏感问题，采用l1,2-范数替代Frobenius范数，该范数对残差列的欧氏范数之和进行惩罚。针对标准与鲁棒情形，分别设计了交替最小化算法和基于IRLS的求解算法，所有算法均被证明收敛至局部极小值。
原文摘要 · Abstract (English)
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equivalence between crisp k-means clustering and matrix factorization, following and rigorously rederiving an unpublished formulation by Bauckhage.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.
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