Title: ExSIS: Extended Sure Independence Screening for Ultrahigh-dimensional Linear Models,
Abstract: Statistical inference can be computationally prohibitive in
ultrahigh-dimensional linear models. Correlation-based variable screening, in
which one leverages marginal correlations for removal of irrelevant variables
from the model prior to statistical inference, can be used to overcome this
challenge. Prior works on correlation-based variable screening either impose
strong statistical priors on the linear model or assume specific post-screening
inference methods. This paper first extends the analysis of correlation-based
variable screening to arbitrary linear models and post-screening inference
techniques. In particular, ($i$) it shows that a condition---termed the
screening condition---is sufficient for successful correlation-based screening
of linear models, and ($ii$) it provides insights into the dependence of
marginal correlation-based screening on different problem parameters. Numerical
experiments confirm that these insights are not mere artifacts of analysis;
rather, they are reflective of the challenges associated with marginal
correlation-based variable screening. Second, the paper explicitly derives the
screening condition for two families of linear models, namely, sub-Gaussian
linear models and arbitrary (random or deterministic) linear models. In the
process, it establishes that---under appropriate conditions---it is possible to
reduce the dimension of an ultrahigh-dimensional, arbitrary linear model to
almost the sample size even when the number of active variables scales almost
linearly with the sample size.