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Robust Manifold Nonnegative Matrix Factorization

Nonnegative Matrix Factorization (NMF) has been one of the most widely used clustering techniques for exploratory data analysis. However, since each data point enters the objective function with squared residue error, a few outliers with large errors easily dominate the objective function. In this article, we propose a Robust Manifold Nonnegative Matrix Factorization (RMNMF) method using ℓ2,1-norm and integrating NMF and spectral clustering under the same clustering framework. We also point out the solution uniqueness issue for the existing NMF methods and propose an additional orthonormal constraint to address this problem. With the new constraint, the conventional auxiliary function approach no longer works. We tackle this difficult optimization problem via a novel Augmented Lagrangian Method (ALM)--based algorithm and convert the original constrained optimization problem on one variable into a multivariate constrained problem. The new objective function then can be decomposed into several subproblems that each has a closed-form solution. More importantly, we reveal the connection of our method with robust K-means and spectral clustering, and we demonstrate its theoretical significance. Extensive experiments have been conducted on nine benchmark datasets, and all empirical results show the effectiveness of our method.