D4R: Doubly robust reduced rank regression in high dimension

Abstract

In this paper, we study high-dimensional reduced rank regression and propose a doubly robust procedure, called $\mathrm{D4R}$, meaning concurrent robustness to both outliers in predictors and heavy-tailed random noise. The proposed method uses the composite gradient descent based algorithm to solve the nonconvex optimization problem resulting from combining Tukey’s biweight loss with spectral regularization. Both theoretical and numerical properties of $\mathrm{D4R}$ are investigated. We establish non-asymptotic estimation error bounds under both the Frobenius norm and the nuclear norm in the high-dimensional setting. Simulation studies and real example show that the performance of $\mathrm{D4R}$ is better than that of several existing estimation methods.