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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 matrix of the loss function, we establish an error bound to the true matrix for those local minima whose ranks are not more than the rank of the true matrix. Then, for the least squares loss function, we achieve the KL property of exponent 1/2 for the F-norm regularized factorization function over its global minimum set under a restricted strong convexity assumption. These theoretical findings are also confirmed by applying an accelerated alternating minimization method to the F-norm regularized factorization problem.

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This paper is concerned with high-dimensional error-in-variables regression that aims at identifying a small number of important interpretable factors for corrupted data from the applications where measurement errors or missing data can not be ignored. Motivated by CoCoLasso due to Datta and Zou (2017) and the advantage of the zeronorm regularized LS estimator over Lasso for clean data, we propose a calibrated zero-norm regularized LS (CaZnRLS) estimator by constructing a calibrated least squares loss with a positive definite projection of an unbiased surrogate for the covariance matrix of covariates, and use the multi-stage convex relaxation approach to compute this estimator. Under a restricted strong convexity on the true covariate matrix, we derive the ℓ 2 -error bound of every iterate and establish the decreasing of the error bound sequence and the sign consistency of the iterates after finite steps. The statistical guarantees are also provided for the CaZnRLS estimator under two types of measurement errors. Numerical comparisons with CoCoLasso and NCL (the nonconvex Lasso of Loh and Wainwright (2012)) show that CaZnRLS has better relative RMSE as well as comparable even more correctly identified predictors.