  Controlling the False Discovery Rate (FDR) in a variable selection procedure
is critical for reproducible discoveries, which received an extensive study in
sparse linear models. However, in many scenarios, such a 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 proposed a
data adaptive FDR control in this structural sparsity setting, the Split
Knockoff method. The proposed scheme relaxes the linear subspace constraint to
its neighborhood, often known as variable splitting in optimization, that
enjoys new statistical benefits. It yields orthogonal design and split knockoff
matrices, that exhibit desired FDR control empirically in structural sparsity
discovery, and improve the power of selecting strong features by enhancing the
incoherence condition for model selection consistency. Yet, the split knockoff
statistics fail to satisfy the exchangeability, a crucial property in the
classical knockoff method for provable FDR control. To address this challenge,
we introduce an almost supermartingale construction under a perturbation of
exchangeability, that enables us to establish FDR control up to an arbitrarily
small inflation that vanishes as the relaxed neighborhood enlarges. Simulation
experiments show the effectiveness of split knockoffs with possible
improvements over knockoffs in both FDR control and Power. An application to
Alzheimer's Disease study with MRI data demonstrates that the split knockoff
method can disclose important lesion regions in brains associated with the
disease and connections between neighboring regions of high contrast variations
during disease progression.
