Enhanced Response Envelope via Envelope Regularization

The response envelope model provides substantial efficiency gains over the standard multivariate linear regression by identifying the material part of the response to the model and by excluding the immaterial part. In this article, we propose the enhanced response envelope by incorporating a novel envelope regularization term based on a nonconvex manifold formulation. It is shown that the enhanced response envelope can yield better prediction risk than the original envelope estimator. The enhanced response envelope naturally handles high-dimensional data for which the original response envelope is not serviceable without necessary remedies. In an asymptotic high-dimensional regime where the ratio of the number of predictors over the number of samples converges to a nonzero constant, we characterize the risk function and reveal an interesting double descent phenomenon for the envelope model. A simulation study confirms our main theoretical findings. Simulations and real data applications demonstrate that the enhanced response envelope does have significantly improved prediction performance over the original envelope method, especially when the number of predictors is close to or moderately larger than the number of samples. Supplementary materials for this article are available online, including a standardized description of the materials available for reproducing the work.