Optimality Conditions And Algorithms For Sparse Principal Component Analysis With L₀-Regularization

Principal component analysis,as an important data dimension reduction tool,has been widely used in various fields of research.However,with the continuous progress of modern science and technology,people are faced with complicated information and need to deal with more and more high dimensions of data.Principal component analysis can no longer meet the needs of The Times in terms of data interpretation,and sometimes even lead to misleading information.In order to improve the inadequacy,sparse principal component analysis emerges at the right moment.It integrates sparsity into principal component analysis,preserves as much original data information as possible,and obtains sparse load vectors.In this paper,the first sparse principal component analysis model with l0-regular term is constructed by adding l0-regular term on the basis of the traditional principal component analysis model.This is a non-convex and nonsmooth sparse optimization problem,which belongs to the NP-hard problem.Then,based on the theoretical development of l0 norm,two kinds of stationary points of the new model are defined: S-stationary point and P-stationary point.Furthermore,the relations between S-stationary point and local optimal solution and between P-stationary point and global optimal solution are deduced,and the theorem is proved.Because of the simple form of the stationary equation,newton algorithm is chosen to solve the first principal component problem.Finally,this paper extends the first sparse principal component model to solve k sparse principal component problem,constructs a new sparse principal component analysis model,and transforms this non-convex and nonsmooth optimization problem with orthogonal constraints into an unconstrained optimization problem on Stiefel manifold.The MADMM algorithm is used to decompose this non-smooth problem into two solvable sub-problems and the framework of the algorithm is given.