Robust graph regularized nonnegative matrix factorization for clustering

Abstract

Nonnegative matrix factorization and its graph regularized extensions have received significant attention in machine learning and data mining. However, existing approaches are sensitive to outliers and noise due to the utilization of the squared loss function in measuring the quality of graph regularization and data reconstruction. In this paper, we present a novel robust graph regularized NMF model (RGNMF) to approximate the data matrix for clustering. Our assumption is that there may exist some entries of the data corrupted arbitrarily, but the corruption is sparse. To address this problem, an error matrix is introduced to capture the sparse corruption. With this sparse outlier matrix, a robust factorization result could be obtained since a much cleaned data could be reconstructed. Moreover, the $\ell _{1}$-norm function is used to alleviate the influence of unreliable regularization which is incurred by unexpected graphs. That is, the sparse error matrix alleviates the impact of noise and outliers, and the $\ell _{1}$-norm function leads to a faithful regularization since the influence of the unreliable regularization errors can be reduced. Thus, RGNMF is robust to unreliable graphs and noisy data. In order to solve the optimization problem of our method, an iterative updating algorithm is proposed and its convergence is also guaranteed theoretically. Experimental results show that the proposed method consistently outperforms many state-of-the-art methods.