  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 paper, 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 non-zero 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. Proofs and additional simulation results are
shown in the supplementary file to this paper.
