Controlling the False Discovery Rate in Transformational Sparsity: Split Knockoffs

Controlling the False Discovery Rate (FDR) in a variable selection procedure is critical for reproducible discoveries, which receives an extensive study in sparse linear models. However, in many scenarios, the sparsity constraint is not directly imposed on the parameters, but on a linear transformation of the parameters to be estimated. Examples can be found in total variations, wavelet transforms, fused LASSO, and trend filtering, etc. In this paper, 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. By variable 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. Moreover, by randomly splitting the data into two independent subsets, new knockoff statistics are generated with signs as independent Bernoulli random variables, enabling inverse supermartingale constructions for provable FDR control. Simulation experiments show that the proposed methodology achieves desired FDR and 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).