  Algorithms that ensure reproducible findings from large-scale,
high-dimensional data are pivotal in numerous signal processing applications.
In recent years, multivariate false discovery rate (FDR) controlling methods
have emerged, providing guarantees even in high-dimensional settings where the
number of variables surpasses the number of samples. However, these methods
often fail to reliably control the FDR in the presence of highly dependent
variable groups, a common characteristic in fields such as genomics and
finance. To tackle this critical issue, we introduce a novel framework that
accounts for general dependency structures. Our proposed dependency-aware T-Rex
selector integrates hierarchical graphical models within the T-Rex framework to
effectively harness the dependency structure among variables. Leveraging
martingale theory, we prove that our variable penalization mechanism ensures
FDR control. We further generalize the FDR-controlling framework by stating and
proving a clear condition necessary for designing both graphical and
non-graphical models that capture dependencies. Additionally, we formulate a
fully integrated optimal calibration algorithm that concurrently determines the
parameters of the graphical model and the T-Rex framework, such that the FDR is
controlled while maximizing the number of selected variables. Numerical
experiments and a breast cancer survival analysis use-case demonstrate that the
proposed method is the only one among the state-of-the-art benchmark methods
that controls the FDR and reliably detects genes that have been previously
identified to be related to breast cancer. An open-source implementation is
available within the R package TRexSelector on CRAN.
