Generalized low rank approximation to the symmetric positive semidefinite matrix

Abstract

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 $n\times k$, 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.