Controlling the False Discovery Rate in Transformational Sparsity: Split Knockoffs

Controlling the False Discovery Rate (FDR) with finite sample guarantee in a variable selection procedure is important for trustworthy and reproducible discoveries, which receives an extensive study in sparse linear models. However, it remains largely open in the scenarios where the sparsity constraint is not directly imposed on the parameters, but on a linear transformation of the parameters to be estimated. Examples include total variations, wavelet transforms, fused LASSO, and trend filtering, etc. In this talk, we propose a data adaptive FDR control in this transformational sparsity setting, the Split Knockoff method. The proposed scheme exploits both variable and data splitting. The linear transformation constraint is relaxed to its Euclidean proximity in a lifted parameter space, yielding an orthogonal design for improved power and orthogonal Split Knockoff copies. To overcome the challenge that exchangeability fails due to the heterogeneous noise brought by the transformation, new inverse supermartingale structures are developed for provable the FDR control with directional effects. Simulation experiments show that the proposed methodology achieves desired (directional) FDR and provides a new opportunity to improve power. An application to Alzheimer's Disease study is provided that atrophy brain regions and their abnormal connections can be discovered based on a structural Magnetic Resonance Imaging dataset (ADNI). This is a joint work with CAO, Yang and SUN, Xinwei.