  In this paper, we investigate the matrix estimation problem in the
multi-response regression model with measurement errors. A nonconvex
error-corrected estimator based on a combination of the amended loss function
and the nuclear norm regularizer is proposed to estimate the matrix parameter.
Then under the (near) low-rank assumption, we analyse statistical and
computational theoretical properties of global solutions of the nonconvex
regularized estimator from a general point of view. In the statistical aspect,
we establish the nonasymptotic recovery bound for any global solution of the
nonconvex estimator, under restricted strong convexity on the loss function. In
the computational aspect, we solve the nonconvex optimization problem via the
proximal gradient method. The algorithm is proved to converge to a near-global
solution and achieve a linear convergence rate. In addition, we also verify
sufficient conditions for the general results to be held, in order to obtain
probabilistic consequences for specific types of measurement errors, including
the additive noise and missing data. Finally, theoretical consequences are
demonstrated by several numerical experiments on corrupted errors-in-variables
multi-response regression models. Simulation results reveal excellent
consistency with our theory under high-dimensional scaling.
