In numerous settings, it is increasingly common to deal with longitudinal data organized as high-dimensional multi-dimensional arrays, also known as tensors. Within this framework, the time-continuous property of longitudinal data often implies a smooth functional structure on one of the tensor modes. To help researchers investigate such data, we introduce a new tensor decomposition approach based on the CANDECOMP/PARAFAC decomposition. Our approach allows for representing a high-dimensional functional tensor as a low-dimensional set of functions and feature matrices. Furthermore, to capture the underlying randomness of the statistical setting more efficiently, we introduce a probabilistic latent model in the decomposition. A covariance-based block-relaxation algorithm is derived to obtain estimates of model parameters. Thanks to the covariance formulation of the solving procedure and thanks to the probabilistic modeling, the method can be used in sparse and irregular sampling schemes, making it applicable in numerous settings. We apply our approach to help characterize multiple neurocognitive scores observed over time in the Alzheimer's Disease Neuroimaging Initiative (ADNI) study. Finally, intensive simulations show a notable advantage of our method in reconstructing tensors.

翻译：在许多研究场景中，处理以高维多维数组（即张量）形式组织的纵向数据日益普遍。在此框架下，纵向数据的时间连续性特性通常意味着张量某一维度上存在平滑的函数结构。为协助研究者分析此类数据，我们提出一种基于CANDECOMP/PARAFAC分解的新型张量分解方法。该方法能够将高维函数张量表示为低维函数集与特征矩阵的组合。此外，为更有效地捕捉统计场景中的潜在随机性，我们在分解中引入了概率潜在模型。通过推导基于协方差的块松弛算法来获取模型参数的估计值。得益于求解过程的协方差公式化表述以及概率建模特性，该方法可适用于稀疏和不规则采样方案，使其在多种场景中具有应用价值。我们将所提方法应用于阿尔茨海默病神经影像倡议（ADNI）研究中随时间观测的多种神经认知评分特征分析。最终，大量仿真实验表明我们的方法在张量重构方面具有显著优势。