  Nonnegative Matrix Factorization (NMF) is a widely-used data analysis
technique, and has yielded impressive results in many real-world tasks.
Generally, existing NMF methods represent each sample with several centroids,
and find the optimal centroids by minimizing the sum of the approximation
errors. However, the outliers deviating from the normal data distribution may
have large residues, and then dominate the objective value seriously. In this
study, an Entropy Minimizing Matrix Factorization framework (EMMF) is developed
to tackle the above problem. Considering that the outliers are usually much
less than the normal samples, a new entropy loss function is established for
matrix factorization, which minimizes the entropy of the residue distribution
and allows a few samples to have large approximation errors. In this way, the
outliers do not affect the approximation of the normal samples. The
multiplicative updating rules for EMMF are also designed, and the convergence
is proved both theoretically and experimentally. In addition, a Graph
regularized version of EMMF (G-EMMF) is also presented to deal with the complex
data structure. Clustering results on various synthetic and real-world datasets
demonstrate the reasonableness of the proposed models, and the effectiveness is
also verified through the comparison with the state-of-the-arts.
