This paper studies the prediction task of tensor-on-tensor regression in which both covariates and responses are multi-dimensional arrays (a.k.a., tensors) across time with arbitrary tensor order and data dimension. Existing methods either focused on linear models without accounting for possibly nonlinear relationships between covariates and responses, or directly employed black-box deep learning algorithms that failed to utilize the inherent tensor structure. In this work, we propose a Factor Augmented Tensor-on-Tensor Neural Network (FATTNN) that integrates tensor factor models into deep neural networks. We begin with summarizing and extracting useful predictive information (represented by the ``factor tensor'') from the complex structured tensor covariates, and then proceed with the prediction task using the estimated factor tensor as input of a temporal convolutional neural network. The proposed methods effectively handle nonlinearity between complex data structures, and improve over traditional statistical models and conventional deep learning approaches in both prediction accuracy and computational cost. By leveraging tensor factor models, our proposed methods exploit the underlying latent factor structure to enhance the prediction, and in the meantime, drastically reduce the data dimensionality that speeds up the computation. The empirical performances of our proposed methods are demonstrated via simulation studies and real-world applications to three public datasets. Numerical results show that our proposed algorithms achieve substantial increases in prediction accuracy and significant reductions in computational time compared to benchmark methods.

翻译：本文研究张量对张量回归的预测任务，其中协变量和响应均为跨时间的多维数组（即张量），且张量阶数和数据维度均可任意。现有方法要么局限于线性模型而未考虑协变量与响应间可能存在的非线性关系，要么直接采用黑箱深度学习算法而未能利用固有的张量结构。本工作提出一种将张量因子模型与深度神经网络相融合的因子增强张量对张量神经网络（FATTNN）。我们首先从具有复杂结构的张量协变量中归纳提取出由"因子张量"表征的有效预测信息，随后以估计的因子张量作为时序卷积神经网络的输入进行预测。所提方法能有效处理复杂数据结构间的非线性关系，在预测精度和计算成本方面均优于传统统计模型与常规深度学习方法。通过运用张量因子模型，本方法既能利用潜在因子结构增强预测性能，又能大幅降低数据维度以加速计算。我们通过仿真研究及在三个公开数据集上的实际应用验证了所提方法的实证性能。数值结果表明，相较于基准方法，本算法在预测精度上实现显著提升，在计算时间上获得大幅缩减。