Generalized low rank approximation to the symmetric positive semidefinite matrix

In this paper, we investigate the generalized low rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $$underset{ rank(X)leq k}{min} sum^m_{i=1}left Vert A_i - B_i XB_i^T right Vert^2_F,$$ where $X$ is an unknown symmetric positive semidefinite matrix and $k$ is a positive integer. We firstly use the property of a symmetric positive semidefinite matrix $X=YY^T$, $Y$ with order $ntimes k$, to convert the generalized low rank approximation into unconstraint generalized optimization problem. Then we apply the nonlinear conjugate gradient method to solve the generalized optimization problem. We give a numerical example to illustrate the numerical algorithm is feasible.