10.6084/m9.figshare.5109382.v2 Jacob Bien Jacob Bien Irina Gaynanova Irina Gaynanova Johannes Lederer Johannes Lederer Christian L. Müller Christian L. Müller Non-Convex Global Minimization and False Discovery Rate Control for the TREX Taylor & Francis Group 2018 Global optimization High-dimensional Model selection Sparsity Tuning parameter 2018-01-25 23:39:23 Journal contribution https://tandf.figshare.com/articles/journal_contribution/Non-convex_Global_Minimization_and_False_Discovery_Rate_Control_for_the_TREX/5109382 <p>The TREX is a recently introduced method for performing sparse high-dimensional regression. Despite its statistical promise as an alternative to the lasso, square-root lasso, and scaled lasso, the TREX is computationally challenging in that it requires solving a nonconvex optimization problem. This article shows a remarkable result: despite the nonconvexity of the TREX problem, there exists a polynomial-time algorithm that is guaranteed to find the global minimum. This result adds the TREX to a very short list of nonconvex optimization problems that can be globally optimized (principal components analysis being a famous example). After deriving and developing this new approach, we demonstrate that (i) the ability of the preexisting TREX heuristic to reach the global minimum is strongly dependent on the difficulty of the underlying statistical problem, (ii) the new polynomial-time algorithm for TREX permits a novel variable ranking and selection scheme, (iii) this scheme can be incorporated into a rule that controls the false discovery rate (FDR) of included features in the model. To achieve this last aim, we provide an extension of the results of Barber and Candes to establish that the <i>knockoff filter</i> framework can be applied to the TREX. This investigation thus provides both a rare case study of a heuristic for nonconvex optimization and a novel way of exploiting nonconvexity for statistical inference.</p>