Orthogonal nonnegative matrix factorization problems for clustering: A new formulation and a competitive algorithm

Orthogonal Nonnegative Matrix Factorization (ONMF) with orthogonality constraints on a matrix has been found to provide better clustering results over existing clustering problems. Because of the orthogonality constraint, this optimization problem is difficult to solve. Many of the existing constraint-preserving methods deal directly with the constraints using different techniques such as matrix decomposition or computing exponential matrices. Here, we propose an alternative formulation of the ONMF problem which converts the orthogonality constraints into non-convex constraints. To handle the non-convex constraints, a penalty function is applied. The penalized problem is a smooth nonlinear programming problem with quadratic (convex) constraints that can be solved by a proper optimization method. We first make use of an optimization method with two gradient projection steps and then apply a post-processing technique to construct a partition of the clustering problem. Comparative performance analysis of our proposed approach with other available clustering methods on randomly generated test problems and hard synthetic data-sets shows the outperformance of our approach, in terms of the obtained misclassification error rate and the Rand index.