We study the asymptotic behavior of a class of methods for sufficient dimension reduction in high-dimension regressions, as the sample size and number of predictors grow in various alignments. It is demonstrated that these methods are consistent in a variety of settings, particularly in abundant regressions where most predictors contribute some information on the response, and oracle rates are possible. Simulation results are presented to support the theoretical conclusion.
"Estimating sufficient reductions of the predictors in abundant high-dimensional regressions." Ann. Statist. 40 (1) 353 - 384, February 2012. https://doi.org/10.1214/11-AOS962