Distributed Proximal Alternating Least Square Method For Tensor Decomposition And Its Application

This paper mainly studies the tensor decomposition algorithm for solving unit constrained optimization problems,and applies it to solve the related problems in chemometrics for measuring the concentration of target substances in a system containing unknown interfering substances.In the 1990 s,Professor Wu Hailong's research group proposed an alternating trilinear decomposition algorithm based on the principle of alternating least squares which can not only avoid the drawbacks of the classic PARAFAC algorithm that are sensitive to rank estimation and slow convergence speed,but also can realize the rapid and accurate determination of multiple target substances in the system where unknown interferers coexist.However,this algorithm does not have the theoretical guarantee of convergence and cannot be established under all conditions.Therefore,in order to solve the existing problems of the algorithm,this paper makes a further study on the nature of tensor decomposition and its practical application in chemometrics,improves the distributed alternating least squares algorithm,adds adjacent terms,and proposes Distributed proximal alternating least squares algorithm.This algorithm theoretically proves the convergence of the algorithm,and it can still achieve good results in the case of inaccurate rank estimation in numerical experiments.In the first chapter,we mainly introduce the development history of tensor decomposition in chemometrics,the current status of research at home and abroad and its research significance,and briefly describe the application of related algorithms in many fields such as biology and chemistry.In the second chapter,we first introduce some basic contents of the tensor used in this article.Then we introduce the alternating trilinear decomposition algorithm,give the mathematical expression of the specific alternating trilinear decomposition algorithm,point out the existing problems in the algorithm,and give specific counterexamples.In the third chapter,we perfected the alternating trilinear decomposition algorithm and obtained the distributed alternating least squares algorithm.However,this algorithm still has the problem that the convergence point is not necessarily a stable point.Therefore,this paper combines the existing methods and regularization techniques,and proposes distributed proximal alternating least squares by adding neighboring terms to solve equation constrained optimization problems.We prove the convergence of the distributed proximal alternating least squares algorithm and give sufficient conditions for the uniqueness of CP decomposition.In the fourth chapter,we conduct real data experiments on the algorithm,analyze and compare the results of the numerical experiments obtained by the algorithm proposed in this paper(distributed adjacent alternating least squares algorithm)and regularized alternating least squares algorithm in the case of inaccurate rank estimation(N = 3,N = 4).