Greedy orthogonal pivoting algorithm for non-negative matrix factorization

Non-negative matrix factorization is a powerful tool for learning useful representations in the data and has been widely applied in many problems such as data mining and signal processing. Orthogonal NMF, which can further improve the locality of decomposition, has drawn considerable interest in clustering problems. However, imposing simultaneous non-negative and orthogonal structure can be difficult, and so existing algorithms can only solve it approximately. To address this challenge, we propose an innovative procedure called Greedy Orthogonal Pivoting Algorithm (GOPA). The GOPA method fully exploits the sparsity of non-negative orthogonal solutions to break the global problem into a series of local optimizations, in which an adaptive subset of coordinates are updated in a greedy, closed-form manner. The biggest advantage of GOPA is that it promotes exact orthogonality and provides solid empirical evidence that stronger orthogonality does contribute favorably to better clustering performance. On the other hand, we have designed randomized and batch-mode version of GOPA, which can further reduce the computational cost and improve accuracy, making it suitable for large data.