On the Degrees of Freedom of Mixed Matrix Regression

With the increasing prominence of big data in modern science, data of interest are more complex and stochastic. To deal with the complex matrix and vector data, this paper focuses on the mixed matrix regression model. We mainly establish the degrees of freedom of the underlying stochastic model, which is one of the important topics to construct adaptive selection criteria for efficiently selecting the optimal model fit. Under some mild conditions, we prove that the degrees of freedom of mixed matrix regression model are the sum of the degrees of freedom of Lasso and regularized matrix regression. Moreover, we establish the degrees of freedom of nuclear-norm regularization multivariate regression. Furthermore, we prove that the estimates of the degrees of freedom of the underlying models process the consistent property.