Rank constrained matrix best approximation problem

In this paper, we study a rank constrained matrix approximation problem in the Frobenius norm: minr(X)=k‖AXB−C‖F2 where kk is a nonnegative integer. First, we derive the feasible interval of the parameter KK for the existence of solutions to the problem. Second, on condition that such a solution exists, we give a general expression for the solution to the corresponding rank constrained matrix approximation problem. Last, we provide the feasible interval of the parameter KK for the existence of the minimal norm of XX.