Group exponential penalized estimation in high-dimensional multivariate linear regression

This paper introduces a penalized least squares method using the group exponential (GEXP) penalty to simultaneously perform variable selection and estimation of the coefficient regression matrix in multivariate linear regression models intended to show the relationship between multiple response variables and multiple explanatory variables. The GEXP penalty is a continuous approximation of the ℓ 0 penalty, which, despite its simple form, avoids the drawbacks of ℓ 0 penalized estimation methods that arise from the discontinuity of the ℓ 0 penalty and thereby provides more stable estimation. We show that the estimator possesses the oracle properties under a high-dimensional asymptotic framework in which the sample size goes to infinity, and both the number of response variables and the number of explanatory variables can go to infinity. We also apply the EGCV criterion as a model selection criterion to choose the two tuning parameters required for the GEXP penalized estimation and show that the EGCV criterion maintains variable selection consistency under certain assumptions. Through numerical experiments, we confirm that the performance of the proposed method is comparable to existing methods.