  Controlling the False Discovery Rate (FDR) in a variable selection procedure
is critical for reproducible discoveries, and it has been extensively studied
in sparse linear models. However, it remains largely open in scenarios where
the sparsity constraint is not directly imposed on the parameters but on a
linear transformation of the parameters to be estimated. Examples of such
scenarios include total variations, wavelet transforms, fused LASSO, and trend
filtering. In this paper, we propose a data-adaptive FDR control method, called
the Split Knockoff method, for this transformational sparsity setting. The
proposed method exploits both variable and data splitting. The linear
transformation constraint is relaxed to its Euclidean proximity in a lifted
parameter space, which yields an orthogonal design that enables the orthogonal
Split Knockoff construction. To overcome the challenge that exchangeability
fails due to the heterogeneous noise brought by the transformation, new inverse
supermartingale structures are developed via data splitting for provable FDR
control without sacrificing power. Simulation experiments demonstrate that the
proposed methodology achieves the desired FDR and power. We also provide an
application to Alzheimer's Disease study, where atrophy brain regions and their
abnormal connections can be discovered based on a structural Magnetic Resonance
Imaging dataset (ADNI).
