Direct decomposition of three-way arrays using a non-negative approximation

Non-negative matrix approximation (NNMA) has been used in diverse scientific fields, but it still has some major limitations. In the present study a novel trilinear decomposition method, termed three-way NNMA (TWNNMA), was developed. The method decomposes three-way arrays directly without unfolding and overcomes the restriction of locking zero elements in the deduced multiplicative update rules by adding a positive symmetric matrix. Direct trilinear decomposition was used as the TWNNMA initialization method and experimental results confirm that this greatly accelerated the convergence. An obvious advantage of TWNNMA is the uniqueness of the non-negative solution, which facilitates a better understanding of the underlying physical realities of complex data. TWNNMA was applied in complex systems such as chemical kinetics, second-order calibration and analysis of GC–MS data. The results demonstrate that TWNNMA, differing from previous trilinear decomposition methods, is comparable to existing second-order calibration methods and represents a promising resolution method for complex systems.