A Bayesian Sparse Kronecker Product Decomposition Framework for Tensor Predictors with Mixed-Type Responses

Abstract

Ultra-high-dimensional imaging tensors are increasingly prevalent in neuroimaging and other biomedical fields. However, few statistical frameworks are capable of jointly modeling continuous, count, and binary outcomes within a unified structure. We propose the Bayesian Sparse Kronecker Product Decomposition (BSKPD), a novel method that represents the regression or classification coefficient tensor as a low-rank sum of Kronecker products of sparse component matrices. This decomposition reshapes both the tensor-valued predictors and coefficients into lower-dimensional matrices, enabling efficient voxel-wise computation via matrix operations while maintaining spatial resolution.

To induce sparsity, we place an element-wise Three-Parameter Beta–Normal (TPBN) shrinkage prior on the component matrices, yielding a parsimonious and interpretable coefficient tensor that highlights truly informative voxels. Our framework adopts a unified exponential-family formulation, accommodating Gaussian, Poisson, and Bernoulli outcomes. By employing Pólya–Gamma augmentation, we derive closed-form Gibbs sampling updates, with computational complexity growing only linearly in the number of voxels.

We establish posterior consistency and identifiability under high-dimensional asymptotics, allowing the image dimensions to grow sub-exponentially with the sample size. This extends Bayesian theory on posterior consistency and identifiability to the setting of mixed-type, multivariate imaging data. Through simulations and real data applications using ADNI and OASIS MRI datasets, we demonstrate that BSKPD offers improved signal localization, enhanced predictive performance, and interpretable scientific insights.