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The non-negative matrix factorization (NMF) model with an additionalorthogonality constraint on one of the factor matrices, called the orthogonalNMF (ONMF), has been found a promising clustering model and can outperform theclassical K-means. However, solving the ONMF model is a challengingoptimization problem because the coupling of the orthogonality andnon-negativity constraints introduces a mixed combinatorial aspect into theproblem due to the determination of the correct status of the variables(positive or zero). Most of the existing methods directly deal with theorthogonality constraint in its original form via various optimizationtechniques, but are not scalable for large-scale problems. In this paper, wepropose a new ONMF based clustering formulation that equivalently transformsthe orthogonality constraint into a set of norm-based non-convex equalityconstraints. We then apply a non-convex penalty (NCP) approach to add them tothe objective as penalty terms, leading to a problem that is efficientlysolvable. One smooth penalty formulation and one non-smooth penalty formulationare respectively studied. We build theoretical conditions for the penalizedproblems to provide feasible stationary solutions to the ONMF based clusteringproblem, as well as proposing efficient algorithms for solving the penalizedproblems of the two NCP methods. Experimental results based on both syntheticand real datasets are presented to show that the proposed NCP methods arecomputationally time efficient, and either match or outperform the existingK-means and ONMF based methods in terms of the clustering performance.

Clustering by Orthogonal NMF Model and Non-Convex Penalty Optimization