  This paper studies the problem of multivariate linear regression where a
portion of the observations is grossly corrupted or is missing, and the
magnitudes and locations of such occurrences are unknown in priori. To deal
with this problem, we propose a new approach by explicitly consider the error
source as well as its sparseness nature. An interesting property of our
approach lies in its ability of allowing individual regression output elements
or tasks to possess their unique noise levels. Moreover, despite working with a
non-smooth optimization problem, our approach still guarantees to converge to
its optimal solution. Experiments on synthetic data demonstrate the
competitiveness of our approach compared with existing multivariate regression
models. In addition, empirically our approach has been validated with very
promising results on two exemplar real-world applications: The first concerns
the prediction of \textit{Big-Five} personality based on user behaviors at
social network sites (SNSs), while the second is 3D human hand pose estimation
from depth images. The implementation of our approach and comparison methods as
well as the involved datasets are made publicly available in support of the
open-source and reproducible research initiatives.
