Title: False Discovery Rate Control via Debiased Lasso,
Abstract: 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 \emph{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 asses its performance and to corroborate our theoretical
results.