Variable screening and spatial smoothing in Fréchet regression with application to diffusion tensor imaging

Modern applications in medical imaging often include high-dimensional predictors and spatially dependent responses in the non-Euclidean space. For example, in imaging-genetics studies, our objective is to study the relationship between single-nucleotide polymorphisms (SNPs), a high-dimensional predictor vector, and diffusion tensor imaging (DTI) responses, which are thousands to millions of voxelwise 3×3 symmetric positive definite (SPD) matrices. In this paper we develop a fast and pragmatic method of regressing spatially associated random responses on a high-dimensional predictor set. Specifically, we focus on two related problems: fast variable screening of high-dimensional predictors and smoothing techniques for non-Euclidean spatially associated responses. Under a Fréchet regression framework (which handles regression of SPD matrix-variate responses on covariates in Euclidean space), we propose a two-stage approach, where a screening method (using distance covariances in metric spaces) is employed to mitigate high-dimensionality (Stage 1), followed by deriving a closed-form solution that powers elegant smoothing of the spatially associated SPD responses (Stage 2). We investigate the finite-sample properties of our method using synthetic data generated under various settings and present illustration via analysis of an imaging-genetics (DTI responses with genetic and demographic predictors) dataset, derived from the Alzheimer’s Disease Neuroimaging Initiative 2. Code for implementing our proposed method is available in the GitHub link: https://github.com/leiyan-ly/Frechet-regression.