False Discovery Rate Control for Grouped Variable Selection in High-Dimensional Linear Models Using the T-Knock Filter

High-dimensional variable selection is a challenging task, especially when groups of highly correlated variables are present in the data, such as in genomics research, direction-of-arrival estimation, and financial engineering. Recently, the T-Knock filter, a new framework for fast variable selection in high-dimensional settings has been developed. It provably controls the false discovery rate (FDR) at a given target level. However, its current version does not consider groups of highly correlated variables, which can lead to a loss in the true positive rate (TPR), i.e., the power. Hence, we propose the T-Knock+GVS filter that allows for grouped variable selection with FDR control in such settings. This is achieved by modifying the forward variable selection algorithm within the T- Knock filter and by adjusting the knockoff generation process such that the generated sets of knockoffs mimic the group correlation structure within the original set of variables. For a special case, we prove that the proposed
$\mathrm{T}-\text{Knock}+\text{GVS}$
filter possesses the grouped variable selection property. Through a simulated high-dimensional genome-wide association study (GWAS), we show that the proposed method significantly increases the TPR, while controlling the FDR at the target level.