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

TL;DR

This paper introduces a Bayesian framework for tensor regression that handles mixed response types, achieves voxel-level sparsity, and scales efficiently to high-dimensional neuroimaging data, improving prediction and interpretability.

Contribution

The paper presents a novel Bayesian sparse Kronecker product decomposition model that unifies mixed-type responses and provides theoretical guarantees in high-dimensional settings.

Findings

Outperforms existing methods in signal recovery and prediction accuracy.

Provides a scalable Gibbs sampling algorithm for high-dimensional tensor data.

Ensures posterior consistency and identifiability in complex models.

Abstract

Ultra-high-dimensional tensor predictors are increasingly common in neuroimaging and other biomedical studies, yet existing methods rarely integrate continuous, count, and binary responses in a single coherent model. We present a Bayesian Sparse Kronecker Product Decomposition (BSKPD) that represents each regression (or classification) coefficient tensor as a low-rank Kronecker product whose factors are endowed with element-wise Three-Parameter Beta-Normal shrinkage priors, yielding voxel-level sparsity and interpretability. Embedding Gaussian, Poisson, and Bernoulli outcomes in a unified exponential-family form, and combining the shrinkage priors with Polya-Gamma data augmentation, gives closed-form Gibbs updates that scale to full-resolution 3-D images. We prove posterior consistency and identifiability even when each tensor mode dimension grows subexponentially with the sample size, thereby extending high-dimensional Bayesian theory to mixed-type multivariate responses. Simulations and applications to ADNI and OASIS magnetic-resonance imaging datasets show that BSKPD delivers sharper signal recovery and lower predictive error than current low-rank or sparsity-only competitors while preserving scientific interpretability.