Third Order Tensor Decomposition Of Nonnegative Unit Constraints In Chemometrics

With the rapid development of science and technology,a large number of tensor data have been produced.Tensor decomposition problems exist widely in various fields,and get the attention of many scientists,such as chemometrics,biology,signal and image processing,blind source separation,statistics and computer networks.It is an important problem in chemometrics to measure the concentration of target substances of interest with unknown interference in complex systems.The quantitative analysis strategy based on mathematical separation is a hot research direction.At present,most of the established models are unconstrained decomposition models,and the alternating trilinear decomposition algorithm(ATLD)is one of the algorithms with fast speed and less memory occupation.However,the convergence theory of the algorithm is relatively lacking,and the practical constraints,such as the non negative requirements of the target concentration,are not considered.According to the constraints of practical problems,this paper proposes a tensor CPdecomposition model with nonnegative unit constraints.Nonnegative constraint means that excitation spectrum(mass spectrum)data,emission spectrum(mass spectrum)data and material concentration data are nonnegative.Unit constraint means that excitation spectrum(mass spectrum)matrix and emission spectrum(mass spectrum)matrix are column unitized.For this model,we propose two algorithms from different perspectives.First,a proximal Gauss-Seidel(PGS)algorithm is proposed to solve the model.By adding a proximal term to ensure sufficient descent,the convergence theory of the algorithm is established.The solution of subproblem is mainly based on fixed point theory.The results of numerical experiments show that the proximal Gauss-Seidel algorithm avoids the negative result in the ATLD algorithm,and is better than the ATLD algorithm in the total average recovery.Secondly,from the point of view of approximate optimization iteration,we want to find the decomposition algorithm of tensor CP with nonnegative unit constraint,and propose the block proximal linear algorithm(BPL).Firstly,a large iterative framework is established based on the idea of block coordinate descent,and the column of decomposition matrix is used as the minimum iterative unit to linearize the subproblem and iterate to the termination condition step by step.Based on the theory of nonconvex optimization,the convergence theory of block proximal linear algorithm is established.Finally,numerical experiments verify the effectiveness of the proposed method,and the numerical results show that the block proximal linear algorithm has more advantages than the ATLD algorithm with post-processing for large-scale problems or the case of adding noise.