An Alternating Feasible Algorithm For Generalized Low Rank Approximation Of Matrices

Recently,the problem of dimensionality reduction and reconstruction has gained wide attentions in pattern recognition and machine learning communities.An important mathematical model of the above problem is the generalized low rank approximations of matrices(GLRAM).The previous literature focuses on the angle of numerical algebra on this problem.In this thesis,we study the algorithm of solving problems from the perspective of the numerical optimization on Stiefel manifold.Combined with the alternating minimization strategy,we propose an alternate feasibility algorithm.We use alternate strategy to solve each orthogonal constraint optimization subproblem.The algorithm use Cayley transform in each iteration steps to determine the update direction.Combined with the curvilinear search and nonmonotone line search with the Barzilai-Borwein step size,we look for the right step in order to make sure every step iteration Armoijo-Wolfe conditions satisfied.In addition,we prove the convergence of subproblems for the gradient descent method with curvilinear search.Numerical examples shows that the proposed algorithm is effective.