  We consider the tuning parameter selection rules for nuclear norm regularized
multivariate linear regression (NMLR) in high-dimensional setting.
High-dimensional multivariate linear regression is widely used in statistics
and machine learning, and regularization technique is commonly applied to deal
with the special structures in high-dimensional data. As we know, how to select
the tuning parameter is an essential issue for regularization approach and it
directly affects the model estimation performance. To the best of our
knowledge, there are no rules about the tuning parameter selection for NMLR
from the point of view of optimization. In order to establish such rules, we
study the duality theory of NMLR. Then, we claim the choice of tuning parameter
for NMLR is based on the sample data and the solution of NMLR dual problem,
which is a projection on a nonempty, closed and convex set. Moreover, based on
the (firm) nonexpansiveness and the idempotence of the projection operator, we
build four tuning parameter selection rules PSR, PSRi, PSRfn and PSR+.
Furthermore, we give a sequence of tuning parameters and the corresponding
intervals for every rule, which states that the rank of the estimation
coefficient matrix is no more than a fixed number for the tuning parameter in
the given interval. The relationships between these rules are also discussed
and PSR+ is the most efficient one to select the tuning parameter. Finally, the
numerical results are reported on simulation and real data, which show that
these four tuning parameter selection rules are valuable.
