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.

翻译：正则化模型已广泛应用于众多领域，特别是在高维数据日益流行的背景下。由于调优参数决定了正则化模型的理论性能与计算效率，其选择问题成为一个基础且重要的议题。本文针对自适应核范数正则化迹回归的调优参数选择问题进行研究，并采用贝叶斯信息准则（BIC）实现。所提出的BIC基于自由度无偏估计量构建，在正则化条件下被证明能够实现调优参数选择的秩一致性——即所选调优参数下的模型解在概率意义上收敛于真实解，且其秩与真实解相同。文中通过数值实验评估了所提BIC在调优参数选择方面的性能。