  The prediction and selection of lesion features are two important tasks in voxel-based neuroimage analysis. Existing multivariate learning models take two tasks equivalently and optimize simultaneously. However, in addition to lesion features, we observe that there is another type of feature, which is commonly introduced during the procedure of preprocessing steps, which can improve the prediction result. We call such a type of feature as procedural bias. Therefore, in this paper, we propose that the features/voxels in neuroimage data are consist of three orthogonal parts: lesion features, procedural bias, and null features. To stably select lesion features and leverage procedural bias into prediction, we propose an iterative algorithm (termed GSplit LBI) as a discretization of differential inclusion of inverse scale space, which is the combination of Variable Splitting scheme and Linearized Bregman Iteration (LBI). Specifically, with a variable the splitting term, two estimators are introduced and split apart, i.e. one is for feature selection (the sparse estimator) and the other is for prediction (the dense estimator). Implemented with Linearized Bregman Iteration (LBI), the solution path of both estimators can be returned with different sparsity levels on the sparse estimator for the selection of lesion features. Besides, the dense the estimator can additionally leverage procedural bias to further improve prediction results. To test the efficacy of our method, we conduct experiments on the simulated study and Alzheimer's Disease Neuroimaging Initiative (ADNI) database. The validity and the benefit of our model can be shown by the improvement of prediction results and the interpretability of visualized procedural bias and lesion features. 
Leveraging both Lesion Features and Procedural Bias in Neuroimaging: An   Dual-Task Split dynamics of inverse scale space