Common psychiatric and brain disorders are highly heritable and affected by a number of genetic risk factors, yet the mechanism by which these genetic factors contribute to the disorders through alterations in brain structure and function remain poorly understood. Contemporary imaging genetic studies integrate genetic and neuroimaging data to investigate how genetic variation contributes to brain disorders via intermediate neuroimaging endophenotypes. However, the large number of potential exposures (genes) and mediators (neuroimaging features) pose new challenges to the traditional mediation analysis. In this paper, we propose a novel multi-exposure-to-multi-mediator mediation model that integrates genetic, neuroimaging and phenotypic data to investigate the "geneneuroimaging-brain disorder" mediation pathway. Our method jointly reduces the dimensions of exposures and mediators into low-dimensional aggregators where the mediation effect is maximized. We further introduce sparsity into the loadings to improve the interpretability. To target the bi-convex optimization problem, we implement an efficient alternating direction method of multipliers algorithm with block coordinate updates. We provide theoretical guarantees for the convergence of our algorithm and establish the asymptotic properties of the resulting estimators. Through extensive simulations, we demonstrate that our method outperforms other competing methods in recovering true loadings and true mediation proportions across a wide range of signal strengths, noise levels, and correlation structures. We further illustrate the utility of the method through a mediation analysis that integrates genetic, brain functional connectivity and smoking behavior data from UK Biobank, and identifies critical genes that impact nicotine dependence via changing the functional connectivity in specific brain regions.

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