Riemannian Manifold Methods For Solving Serval Matrix Trace Function Minimization Problems

Matrix trace function minimization problem is a kind of constraint matrix optimization problem in essence which satisfies certain constraint conditions can be regarded as a variable matrix trace function minimization and matrix or its special form as the objective function of optimization problem.It has been used in machine learning,principal component analysis,feature extraction,image processing and other subjects widely.Because the actual background is different,the constraint condition of unknown variable or the form of matrix trace function is different,which puts forward many different matrix trace function minimization problems,and because the determined matrix set is different at the same time,the properties of matrix class are different,so the corresponding solving method technique and difficulty are different.This thesis mainly studies some matrix trace function minimization problems under the product manifold.The details are as follows.In chapter 2,we study a class of matrix trace function minimization problem derived from feature extraction,which can be transformed into a class of matrix trace function minimization problem constrained by Stiefel manifolds and then design Riemannian Newton method to solve the problem.In particular,the resultant Newton's equation can be technically transformed into a standard real symmetric linear system by means of Kronecker product and vectorization operators.The Riemannian gradient method is combined with the Newton's method to produce a hybrid algorithm,which is globally and quadratically convergent in practical computations.Some numerical tests and numerical comparisons are given to demonstrate the efficiency of the proposed method.In chapter 3,we study the trace function minimization problem of a class of complex Stiefel manifold constrained matrix with generalized eigenvalue minimum perturbation problem derived from non-square constraints,based on the Riemannian gradient of the objective function and the Riemannian Hessain matrix.In this chapter,the Riemannian trust region method is designed to solve the problem.Combined with the transformation form of Riemannian Hessian in the tangent space of the product manifold,the original trust region subproblem is transformed into a new subproblem containing simple linear constraints and norm constraints,and the classical truncated conjugate gradient method is used to solve the problem.Sufficient numerical comparisons which includs the first order Riemannian algorithm and several classes of recent infeasible methods,demonstrate the effectiveness of the proposed algorithm.In chapter 4,we study a matrix trace function minimization problem derived from the Individual Difference Scale(INDSCAL)model with manifold and linear constraints.Based on Modified in European space Polak-Ribi`ere-Polyak conjugate gradient method(MPRP),the design is suitable for the problem of model algorithm to solve the Riemannian MPRP conjugate gradient method,and combined with complete global convergence analysis of the numerical experiment is given to illustrate the effectiveness of the algorithm.