In multiple-output regression model with input variables lying in a high-dimensional space, inverse-covariance regularized sparse multivariate regression has been widely applied to obtain a sparse set of parameter estimates. In the standard formulation of this problem, the outputs are assumed to be fully observed. However, this assumption is violated in many applications. For example, in the field of genomics, it is common for sequencing-based measurements to be missing at some genomic positions due to the stochasticity in capturing. Building on error-in-variables concepts, we develop a method for sparse high-dimensional multivariate regression with contaminated outputs by using designed unbiased surrogates. This approach iterates over two convex sub-problems, leading to an improvement in computational efficiency compared to existing non-convex techniques. Preliminary results from simulations indicate that our method enjoys high efficiency and robustness across different scenarios.