An inexactly accelerated algorithm for nonnegative tensor CP decomposition with the column unit constraints

The component separation problem in complex chemical systems is very important and challenging in chemometrics. In this paper, we study a third-order nonnegative CANDECOMP/PARAFAC decomposition model with the column unit constraints (NCPD_CU) motivated by the component separation problem. To solve the NCPD_CU model, we first explore rapid computational methods for a generalized class of three-block optimization problems, which may exhibit nonconvexity and nonsmoothness. To this end, we propose the accelerated inexact block coordinate descent (AIBCD) algorithm, where each subproblem is inexactly solved through a finite number of inner-iterations employing the alternating proximal gradient method. Additionally, the algorithm incorporates extrapolation during the outer-iterations to enhance overall efficiency. We prove that the iterative sequence generated by the algorithm converges to a stationary point under mild conditions. Subsequently, we apply this methodology to the NCPD_CU model that satisfies the specified conditions. Finally, we present numerical results using both synthetic and real-world data, showcasing the remarkable efficiency of our proposed method.