We propose a novel tensor-on-tensor modeling framework that flexibly models nonlinear voxel-level relationships using Gaussian process (GP) priors, while incorporating the spatial structure of the output tensor through low-rank tensor-based coefficients. Spatial heterogeneity is captured through patch-to-voxel mappings, enabling each output voxel to depend on its spatial neighborhood. The proposed interpretable and flexible Bayesian tensor-on-tensor framework is able to capture nonlinearity, spatial information, and spatial heterogeneity. We develop an efficient Markov chain Monte Carlo (MCMC) algorithm that exploits parallel structure to sample voxel-specific GP atoms and update low-rank tensor coefficients. Extensive simulations reveal advantages of the proposed approach over existing methods in terms of coefficient estimation, inference, prediction, and scalability to high-dimensional images. Applied to longitudinal image prediction with T1-weighted MRIs from the Alzheimer's Disease Neuroimaging Initiative (ADNI), the proposed method can accurately forecast future cortical thickness. The predicted images also enable reliable prediction of brain aging, underscoring their biological relevance. Overall, the ADNI analysis highlights the model's ability to forecast future neurobiological changes that has important implications for early detection of AD.

翻译：摘要：我们提出了一种新颖的张量对张量建模框架，该框架利用高斯过程先验灵活建模非线性体素级关系，同时通过低秩张量系数融合输出张量的空间结构。通过补丁到体素的映射捕捉空间异质性，使每个输出体素能够依赖其空间邻域。所提出的可解释且灵活的贝叶斯张量对张量框架能够捕获非线性、空间信息和空间异质性。我们开发了一种高效的马尔可夫链蒙特卡洛算法，利用并行结构采样体素特定的高斯过程原子并更新低秩张量系数。大量仿真表明，该方法在系数估计、推断、预测以及对高维图像的可扩展性方面优于现有方法。基于阿尔茨海默病神经影像倡议的T1加权MRI进行纵向图像预测的应用中，所提方法能准确预测未来皮层厚度。预测图像同时能可靠预测大脑老化，凸显其生物学相关性。总体而言，ADNI分析强调了该模型预测未来神经生物学变化的能力，对AD早期检测具有重要意义。