On the consistency theory of high dimensional variable screening

Variable screening is a fast dimension reduction technique for assisting high dimensional feature selection. As a preselection method, it selects a moderate size subset of candidate variables for further refining via feature selection to produce the final model. The performance of variable screening depends on both computational efficiency and the ability to dramatically reduce the number of variables without discarding the important ones. When the data dimension $p$ is substantially larger than the sample size $n$, variable screening becomes crucial as 1) Faster feature selection algorithms are needed; 2) Conditions guaranteeing selection consistency might fail to hold. This article studies a class of linear screening methods and establishes consistency theory for this special class. In particular, we prove the restricted diagonally dominant (RDD) condition is a necessary and sufficient condition for strong screening consistency. As concrete examples, we show two screening methods $SIS$ and $HOLP$ are both strong screening consistent (subject to additional constraints) with large probability if $n > O((\rho s + \sigma/\tau)^2\log p)$ under random designs. In addition, we relate the RDD condition to the irrepresentable condition, and highlight limitations of $SIS$.
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