Envelopes and principal component regression

TL;DR

This paper introduces NIECE, a non-iterative, computationally efficient envelope estimation method for high-dimensional regression models, bridging envelope techniques with principal component regression to enhance estimation accuracy.

Contribution

The paper proposes a novel non-iterative envelope estimation method, NIECE, that simplifies and accelerates envelope modeling in high-dimensional settings, connecting it with principal components regression.

Findings

NIECE outperforms iterative methods in high-dimensional simulations.

Theoretical bounds relate envelope estimation error to eigenvalue gaps.

Application to various models shows significant improvement over standard methods.

Abstract

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.