Semi-Orthogonal Non-Negative Matrix Factorization
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Non-negative Matrix Factorization (NMF) is a popular clustering and dimension reduction method by decomposing a non-negative matrix into the product of two lower dimension matrices composed of basis vectors. In this paper, we propose a semi-orthogonal NMF method that enforces one of the matrices to be orthogonal with mixed signs, thereby guarantees the rank of the factorization. Our method preserves strict orthogonality by implementing the Cayley transformation to force the solution path to be exactly on the Stiefel manifold, as opposed to the approximated orthogonality solutions in existing literature. We apply a line search update scheme along with an SVD-based initialization which produces a rapid convergence of the algorithm compared to other existing approaches. In addition, we present formulations of our method to incorporate both continuous and binary design matrices. Through various simulation studies, we show that our model has an advantage over other NMF variations regarding the accuracy of the factorization, rate of convergence, and the degree of orthogonality while being computationally competitive. We also apply our method to a text-mining data on classifying triage notes, and show the effectiveness of our model in reducing classification error compared to the conventional bag-of-words model and other alternative matrix factorization approaches.
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