| Multivariate linear regression model is widely applied in many fields such as econometrics and genetic engineering.Reduced rank regression offers an effective way for dimension reduction,which helps to better model the relationship between response and predictor variables.Reduced rank regression methods achieve dimension reduction by restricting the low-rank structure of the coefficient matrix.The low-rank constraint of the coefficient matrix facilitates parameter estimation and improves the predictive accuracy of the model.Based on the low-rank structure,many sparse reduced rank regression methods introduce different sparse encouraging penalties on the coefficient matrix to promote model interpretation and achieve variable selection.However,many sparse reduced-rank regression methods are very sensitive to outliers.The low-rank sparse structure of the coefficient matrix can be easily distorted by outliers.In practical applications,outliers are often unavoidable.Therefore,it is very important to establish robust sparse reduced rank regression methods.Based on the multivariate mean-shift regression model,we propose a sparse reduced rank regression approach to achieve low-rank sparse estimation and outlier detection simultaneously.The mean-shift matrix in the model can represent any possible combination of outliers,which can be estimated to indicate possible outliers.The rank constraint and the Group-Lasso type penalty for the coefficient matrix encourage the low-rank row sparse structure of coefficient matrix and help to achieve dimension reduction and variable selection.When choosing an appropriate penalty function for the mean-shift matrix,the square loss in our optimization objective function can be related to a robust loss,thus the robustness of the method can be guaranteed.We perform a simple non-asymptotic analysis for the proposed method and an iterative algorithm is developed for solving the proposed problem.Different models are designed for simulation and finally the proposed method is applied to real data applications.The simulation results and real data analysis show that the proposed method can effectively identify outliers and improve the accuracy of model prediction and variable selection in the presence of outliers. |
