  We consider the problem of variable selection in high-dimensional statistical
models where the goal is to report a set of variables, out of many predictors
$X_1, \dotsc, X_p$, that are relevant to a response of interest. For linear
high-dimensional model, where the number of parameters exceeds the number of
samples $(p>n)$, we propose a procedure for variables selection and prove that
it controls the "directional" false discovery rate (FDR) below a pre-assigned
significance level $q\in [0,1]$. We further analyze the statistical power of
our framework and show that for designs with subgaussian rows and a common
precision matrix $\Omega\in\mathbb{R}^{p\times p}$, if the minimum nonzero
parameter $\theta_{\min}$ satisfies $$\sqrt{n} \theta_{\min} - \sigma
\sqrt{2(\max_{i\in [p]}\Omega_{ii})\log\left(\frac{2p}{qs_0}\right)} \to
\infty\,,$$ then this procedure achieves asymptotic power one. Our framework is
built upon the debiasing approach and assumes the standard condition $s_0 =
o(\sqrt{n}/(\log p)^2)$, where $s_0$ indicates the number of true positives
among the $p$ features. Notably, this framework achieves exact directional FDR
control without any assumption on the amplitude of unknown regression
parameters, and does not require any knowledge of the distribution of
covariates or the noise level. We test our method in synthetic and real data
experiments to assess its performance and to corroborate our theoretical
results.
