Error bound of critical points and KL property of exponent $1/2$ for squared F-norm regularized factorization
This provides theoretical guarantees for factorization methods in matrix recovery, but it is incremental as it builds on existing regularization techniques.
The paper tackles noisy low-rank matrix recovery by analyzing squared F-norm regularized factorization, deriving error bounds for critical points and establishing KL properties with exponent 1/2 under various sampling settings.
This paper is concerned with the squared F(robenius)-norm regularized factorization form for noisy low-rank matrix recovery problems. Under a suitable assumption on the restricted condition number of the Hessian for the loss function, we derive an error bound to the true matrix for the non-strict critical points with rank not more than that of the true matrix. Then, for the squared F-norm regularized factorized least squares loss function, under the noisy and full sample setting we establish its KL property of exponent $1/2$ on its global minimizer set, and under the noisy and partial sample setting achieve this property for a class of critical points. These theoretical findings are also confirmed by solving the squared F-norm regularized factorization problem with an accelerated alternating minimization method.