Application of non-linear optimization for estimating Tucker3 solutions

Abstract An extension of the non-linear optimization (FMIN) which was previously proposed to the estimation of multivariate curve resolution solutions by Tauler [AnalyticaChimicaActa 595 (2007) 289–298] to the multi-way analysis is reported in this contribution. TuckFMIN is presented for the calculation of Tucker3 solutions by the FMIN. TuckFMIN is heavily relying on the minimization of an objective function which is defined directly from the constraints non-fulfillment. Starting from higher-order singular value decomposition (HO-SVD) loadings, TuckFMIN represents a new approach for obtaining three proper rotation matrices for transforming the HO-SVD loadings to physically and chemically meaningful solutions. Different constraints should be imposed during optimization; sparsity and trilinearity are among new constraints for simplicity of the Tucker3 core and fast and robust convergence in a multi-way decomposition. Simulated fluorescence data was exemplified to evaluate the feasibility of proposed method. For the sake of comparison between different initialization methods, PARAFAC-ALS and HO-SVD loadings were used. LOF (lack of fit) from TuckFMIN modeling is always as same as LOF of HO-SVD and less than LOF of the PARAFAC-ALS for noise free data sets. Despite the PARAFAC-ALS decomposition, TuckFMIN has the best fitting regardless to rank-deficiency or levels of noise. By means of simulation study, it is demonstrated that TuckFMIN can be helpful for faster convergence and obtaining the reproducible results. An experimental 3D fluorescence data set from gold nano-particle (AuNP) interaction with HIV genome are successfully used for evaluating the performance of the TuckFMIN algorithm.