Calmness of partial perturbation to composite rank constraint systems and its applications

This paper is concerned with the calmness of a partial perturbation to the composite rank constraint system, an intersection of the rank constraint set and a general closed set, which is shown to be equivalent to a local Lipschitz-type error bound and also a global Lipschitz-type error bound under a certain compactness. Based on its lifted formulation, we derive two criteria for identifying those closed sets such that the associated partial perturbation possesses the calmness, and provide a collection of examples to demonstrate that the criteria are satisfied by common nonnegative and positive semidefinite rank constraint sets. Then, we use the calmness of this perturbation to obtain several global exact penalties for rank constrained optimization problems, and a family of equivalent DC surrogates for rank regularized problems

KL property of exponent 1/21/2 of ℓ2,0\ell_{2,0}-norm and DC regularized factorizations for low-rank matrix recovery

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.Comment: 29 pages, 3 figure