KL property of exponent $1/2$ of $\ell_{2,0}$-norm and DC regularized factorizations for low-rank matrix recovery
This work addresses low-rank matrix recovery for applications like signal processing or machine learning, but it is incremental as it builds on existing factorization methods with specific theoretical refinements.
The paper tackles the problem of low-rank matrix recovery when only a coarse rank estimate is available, by adding an ℓ₂,₀-norm regularized term to a factored loss function and introducing a balanced term to handle factorization ambiguities, establishing the Kurdyka-Łojasiewicz (KL) property with exponent 1/2 for the nonsmooth function and its equivalent difference-of-convex (DC) reformulations under a restricted condition number assumption.
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.