Abstract
Multivariate regression models are widely used in various fields for fitting multiple responses. In this paper, we proposed a sparse Laplacian shrinkage estimator for the high-dimensional multivariate regression models. We consider two graphical structures among predictors and responses. The proposed method explores the regression relationship allowing the predictors and responses derived from different multivariate normal distributions with general covariance matrices. In practice, the correlations within data are often complex and interact with each other based on the regression function. The proposed method solves this problem by building a structured penalty to encourage the shared structure between the graphs and the regression coefficients. We provide theoretical results under reasonable conditions and discuss the related algorithm. The effectiveness of the proposed method is demonstrated in a variety of simulations as well as an application to the index tracking problem in the stock market.
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