A Bayesian Sparse Kronecker Product Decomposition Framework for Tensor Predictors With Mixed‐Type Responses: Applications to Neuroimaging Data Mining

Abstract

High‐dimensional tensor‐valued predictors are now routine in neuroimaging and other data‐rich domains, yet few statistical frameworks can jointly analyze continuous, binary, and count outcomes while scaling to realistic image resolutions. We propose the Bayesian sparse Kronecker product decomposition (BSKPD), which represents a regression or classification coefficient tensor as a low‐rank sum of Kronecker products of sparse component tensors. A sparse Kronecker product decomposition transformation reshapes tensor predictors and coefficients into lower‐dimensional matrices, enabling voxel‐level computation through standard matrix operations while preserving spatial structure. Sparsity is induced by a three‐parameter Beta–Normal (TPBN) global–local shrinkage prior on the Kronecker factors, yielding parsimonious and interpretable coefficient tensors that highlight informative brain regions. A unified exponential‐family formulation accommodates Gaussian, Bernoulli, and negative‐binomial responses, and Pólya–Gamma augmentation leads to closed‐form Gibbs updates. We establish identifiability and posterior consistency in both classical and high‐dimensional regimes, extending Bayesian theory to mixed‐type multivariate tensor regression. Simulations and Alzheimer's disease neuroimaging applications show that BSKPD yields interpretable whole‐brain coefficient maps while maintaining competitive predictive accuracy relative to existing approaches.