  This paper is concerned with the factorization form of the rank regularized
loss minimization problem. To cater for the scenario in which only a coarse
estimation is available for the rank of the true matrix, an $\ell_{2,0}$-norm
regularized term is added to the factored loss function to reduce the rank
adaptively; and account for the ambiguities in the factorization, a balanced
term is then introduced. For the least squares loss, under a restricted
condition number assumption on the sampling operator, we establish the KL
property of exponent $1/2$ of the nonsmooth factored composite function and its
equivalent DC reformulations in the set of their global minimizers. We also
confirm the theoretical findings by applying a proximal linearized alternating
minimization method to the regularized factorizations.
