This paper is concerned with the model of column ${\ell }_{2,0}$-regularized factorization for inductive matrix completion. The column ${\ell }_{2,0}$-norm is an exact variational characterization of the rank function, promoting low-rank structure through column sparsity of the matrix. We established that the stationary points between the ${\ell }_{2,0}$-norm regularized factorization model and the rank regularized model are equivalent. Under a suitable assumption, an error bound to the true matrix was obtained for the stationary points with rank not more than that of the true matrix. For this nonconvex discontinuous optimization problem, we developed a proximal alternating minimization (PAM) algorithm with subspace correction and showed that the whole sequence globally converges to a stationary point of the rank regularized model. Numerical experiments conducted with examples of both synthetic and real data demonstrated the effectiveness of the proposed method.