Adaptive Graph Regularized Low–Rank Matrix Factorization With Noise and Outliers for Clustering

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Clustering, which is a commonly used tool, has been applied in machine learning, data mining and so on, and has received extensive research. However, there are usually noise and outliers in the data, which will bring about significant errors in the clustering results. In this paper, a robust clustering model with adaptive graph regularization (RCAG) is proposed, on which, sparse error matrix is introduced to express sparse noise, such as impulse noise, dead line, stripes, and $\ell _{1}$ norm is introduced to alleviate the sparse noise. In addition, the $\ell _{2,1}$ norm is also proposed mitigating the effects of outliers, and it has rotation invariance property. Therefore, our RCAG is insensitive to data noise and outliers. More importantly, the adaptive graph regularization is introduced into the RCAG to improve the clustering performance. Aiming at the optimization objective, we propose an iterative updating algorithm, named the Augmented Lagrangian Method (ALM), to update each optimization variable respectively. The convergence and time complexity of RCAG is also proved in theory. Finally, experimental results on fourteen datasets of four application scenarios, such as face image, handwriting recognition and UCI, elaborate the superiority of proposed method over seven existing classical clustering methods. The experimental results demonstrate that our approach achieves better clustering performance in ACC and Purity, which is a little less impressive in other ways.