Bayesian Generalized Sparse Symmetric Tensor-on-Vector Regression

Motivated by brain connectome datasets acquired using diffusion weighted magnetic resonance imaging (DWI), this article proposes a novel generalized Bayesian linear modeling framework with a symmetric tensor response and scalar predictors. The symmetric tensor coefficients corresponding to the scalar predictors are embedded with two features: low-rankness and group sparsity within the low-rank structure. Besides offering computational efficiency and parsimony, these two features enable identification of important “tensor nodes” and “tensor cells” significantly associated with the predictors, with characterization of uncertainty. The proposed framework is empirically investigated under various simulation settings and with a real brain connectome dataset. Theoretically, we establish that the posterior predictive density from the proposed model is “close” to the true data generating density, the closeness being measured by the Hellinger distance between these two densities, which scales at a rate very close to the finite dimensional optimal rate of n−1/2, depending on how the number of tensor nodes grow with the sample size. The theoretical results with proofs are provided in the supplementary materials which are available online.