Generalized low-rank approximation to the symmetric positive semidefinite matrix

In this paper, we consider the generalized low-rank approximation to the symmetric positive semidefinite matrix in the Frobenius norm: $ \underset{X}{\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 whose rank is less than or equal to a positive integer $ k $. We first characterize the feasible set as $ X = YY^{T} $, where $ Y $ has the order $ n\times k $, and then convert the generalized low-rank approximation into an unconstrained generalized optimization problem. Finally, we employ the nonlinear conjugate gradient method with an exact line search to solve the generalized optimization problem. We also give numerical examples to exemplify the results.