Theories And Numerical Algorithms For Some Constrained Matrix Minimization Problems

The constrained matrix minimization problem is one of the most important problems in matrix theory and its applications.It is widely used in image processing,pattern recognition,signal processing,semantic recognition and other fields.This paper systematically studies the following three types of constrained matrix minimization problem of transformation.In Chapter2,the matrix minimization problem of image salient object detection is studied.the matrices L∈Rm×n,S∈Rm×n are the background feature matrix and the salient object feature matrix of the image feature matrix respectively.First,the sparse regularization technique is used to process the Schatten-p norm,the original problem is reconstructed into an equivalent matrix minimization problem,and then the reconstructed problem is solved by using the alternating direction multiplier method.The convergence analysis is given.Finally,numerical experiments show that the feasibility and effectiveness of the new algorithm.In Chapter 3,we study the matrix minimization problem in self-representing salient object detection among them,the matrix L∈Rn×n is the coefficient matrix,S∈Rm×n is the background feature matrix.First,a matrix minimization model for self-expressive salient object detection is established,and the problem is transformed into a convex minimization problem,then the fixed point iteration algorithm is used to solve the problem.After that,the convergence theorem is given.Finally,we performed numerical experiments on multiple datasets.Compared with traditional algorithms,the new algorithm has higher coverage and accuracy of salient object detection.In Chapter 4,we study the matrix minimization problems in deformable models where P ∈ Rn×m is the force matrix in the deformable model,and Q∈Rn×m is the observed object deformation matrix.Firstly,a mapping from an arbitrary square matrix to nonsymmetric semidefinite cone is given.Based on the idea of the proximal gradient descent method,a projection soft threshold iteration algorithm and an acceleration strategy are designed to solve the problem,and then the convergence of the algorithm is analyzed.The effectiveness of the algorithm is verified by numerical experiments.