Motivated from the study of l_1-regularization with a sparsity constraint in compressed sensing, we investigate the theoretical properties of nuclear norm regularization with a low-rank constraint for matrix completion in this paper. Two types of regularization methods have been studied for matrix completion: the residual method and the Tikhonov method. We propose and discuss a group of regularization conditions under which the residual method provides regularization. Moreover, we investigate the Tikhonov regularization under some source and restricted injective conditions and derive the stability of the minimizer, as well as its consistency and convergence rates.
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