  Regularized models have been applied in lots of areas, with high-dimensional
data sets being popular. Because tuning parameter decides the theoretical
performance and computational efficiency of the regularized models, tuning
parameter selection is a basic and important issue. We consider the tuning
parameter selection for adaptive nuclear norm regularized trace regression,
which achieves by the Bayesian information criterion (BIC). The proposed BIC is
established with the help of an unbiased estimator of degrees of freedom. Under
some regularized conditions, this BIC is proved to achieve the rank consistency
of the tuning parameter selection. That is the model solution under selected
tuning parameter converges to the true solution and has the same rank with that
of the true solution in probability. Some numerical results are presented to
evaluate the performance of the proposed BIC on tuning parameter selection.
