  Envelope methods offer targeted dimension reduction for various models. The
overarching goal is to improve efficiency in multivariate parameter estimation
by projecting the data onto a lower-dimensional subspace known as the envelope.
Envelope approaches have advantages in analyzing data with highly correlated
variables, but their iterative Grassmannian optimization algorithms do not
scale very well with ultra high-dimensional data. While the connections between
envelopes and partial least squares in multivariate linear regression have
promoted recent progress in high-dimensional studies of envelopes, we propose a
more straightforward way of envelope modeling from a novel principal components
regression perspective. The proposed procedure, Non-Iterative Envelope
Component Estimation (NIECE), has excellent computational advantages over the
iterative Grassmannian optimization alternatives in high dimensions. We develop
a unified NIECE theory that bridges the gap between envelope methods and
principal components in regression. The new theoretical insights also shed
light on the envelope subspace estimation error as a function of eigenvalue
gaps of two symmetric positive definite matrices used in envelope modeling. We
apply the new theory and algorithm to several envelope models, including
response and predictor reduction in multivariate linear models, logistic
regression, and Cox proportional hazard model. Simulations and illustrative
data analysis show the potential for NIECE to improve standard methods in
linear and generalized linear models significantly.
