Brain imaging genetics studies the genetic basis of brain structures and functions via integrating both genotypic data such as single nucleotide polymorphism (SNP) and imaging quantitative traits (QTs). In this area, both multi-task learning (MTL) and sparse canonical correlation analysis (SCCA) methods are widely used since they are superior to those independent and pairwise univariate analyses. MTL methods generally incorporate a few of QTs and are not designed for feature selection from a large number of QTs; while existing SCCA methods typically employ only one modality of QTs to study its association with SNPs. Both MTL and SCCA encounter computational challenges as the number of SNPs increases. In this paper, combining the merits of MTL and SCCA, we propose a novel multi-task SCCA (MTSCCA) learning framework to identify bi-multivariate associations between SNPs and multi-modal imaging QTs. MTSCCA could make use of the complementary information carried by different imaging modalities. Using the G _2,1-norm regularization, MTSCCA treats all SNPs in the same group together to enforce sparsity at the group level. The l 2 , 1 -norm penalty is used to jointly select features across multiple tasks for SNPs, and across multiple modalities for QTs. A fast optimization algorithm is proposed using the grouping information of SNPs. Compared with conventional SCCA methods, MTSCCA obtains improved performance regarding both correlation coefficients and canonical weights patterns. In addition, our method runs very fast and is easy-to-implement, and thus could provide a powerful tool for genome-wide brain-wide imaging genetic studies.

Brain imaging genetics intends to uncover associations between genetic markers and neuroimaging quantitative traits. Sparse canonical correlation analysis (SCCA) can discover bi-multivariate associations and select relevant features, and is becoming popular in imaging genetic studies. The L1-norm function is not only convex, but also singular at the origin, which is a necessary condition for sparsity. Thus most SCCA methods impose [Formula: see text]-norm onto the individual feature or the structure level of features to pursuit corresponding sparsity. However, the [Formula: see text]-norm penalty over-penalizes large coefficients and may incurs estimation bias. A number of non-convex penalties are proposed to reduce the estimation bias in regression tasks. But using them in SCCA remains largely unexplored. In this paper, we design a unified non-convex SCCA model, based on seven non-convex functions, for unbiased estimation and stable feature selection simultaneously. We also propose an efficient optimization algorithm. The proposed method obtains both higher correlation coefficients and better canonical loading patterns. Specifically, these SCCA methods with non-convex penalties discover a strong association between the APOE e4 rs429358 SNP and the hippocampus region of the brain. They both are Alzheimer's disease related biomarkers, indicating the potential and power of the non-convex methods in brain imaging genetics.