Low rank approximation of the symmetric positive semidefinite matrix

In this paper, we consider the low rank approximation of the symmetric positive semidefinite matrix, which arises in machine learning, quantum chemistry and inverse problem. We first characterize the feasible set by X=YYT,Y∈Rn×kX=YYT,Y∈Rn×k, and then transform low rank approximation into an unconstrained optimization problem. Finally, we use the nonlinear conjugate gradient method with exact line search to compute the optimal low rank symmetric positive semidefinite approximation of the given matrix. Numerical examples show that the new method is feasible and effective.