Adaptive Huber trace regression with low-rank matrix parameter via nonconvex regularization

Abstract

In this paper, we consider the adaptive Huber trace regression model with matrix covariates. A non-convex penalty function is employed to account for the low-rank structure of the unknown parameter. Under some mild conditions, we establish an upper bound for the statistical rate of convergence of the regularized matrix estimator. Theoretically, we can deal with heavy-tailed distributions with bounded $(1+\delta )$-th moment for any $\delta >0$. Furthermore, we derive the effect of the adaptive parameter on the final estimator. Some simulations, as well as a real data example, are designed to show the finite sample performance of the proposed method.