  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 many applications where measurement errors or missing data
can not be ignored. Motivated by CoCoLasso due to Datta and Zou \cite{Datta16}
and the advantage of the zero-norm 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 the
CaZnRLS estimator. Under a restricted eigenvalue condition on the true matrix
of covariates, we derive the $\ell_2$-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 proposed by
Poh and Wainwright \cite{Loh11}) demonstrate that CaZnRLS not only has the
comparable or even better relative RSME but also has the least number of
incorrect predictors identified.
