Low-rank matrix estimation in multi-response regression with measurement errors: Statistical and computational guarantees

In this paper, we investigate the matrix estimation problem in multi-response regression with measurement errors. A nonconvex error-corrected estimator is proposed to estimate the matrix parameter via a combination of the loss function and the nuclear norm regularization. Then under the low-rank constraint, we analyse the statistical and computational theoretical properties of global solution of the nonconvex regularized estimator from a general point. In the statistical aspect, we establish the recovery bound for the 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 establish sufficient conditions for the general results to be held for specific types of corruptions, including the additive noise and missing data. Probabilistic consequences are obtained by applying the general results. Finally, we demonstrate our theoretical consequences by several numerical experiments on the corrupted errors-in-variables multi-response regression models. Simulation results show remarkable consistency with our theory under high-dimensional scaling.