Robust sparse principal component regression under the high dimensional elliptical model

In this paper we focus on the principal component regression and its application to high dimension non-Gaussian data. The major contributions are two folds. First, in low dimensions and under the Gaussian model, by borrowing the strength from recent development in minimax optimal principal component estimation, we first time sharply characterize the potential advantage of classical principal component regression over least square estimation. Secondly, we propose and analyze a new robust sparse principal component regression on high dimensional elliptically distributed data. The elliptical distribution is a semiparametric generalization of the Gaussian, including many well known distributions such as multivariate Gaussian, rank-deficient Gaussian, t, Cauchy, and logistic. It allows the random vector to be heavy tailed and have tail dependence. These extra flexibilities make it very suitable for modeling finance and biomedical imaging data. Under the elliptical model, we prove that our method can estimate the regression coefficients in the optimal parametric rate and therefore is a good alternative to the Gaussian based methods. Experiments on synthetic and real world data are conducted to illustrate the empirical usefulness of the proposed method.