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.

通过提取和总结有用的预测信息（由 “因子张量” 表示）并利用张量分解模型将其输入到时间卷积神经网络中，本研究提出了一种因子增强的张量对张量神经网络（FATTNN）。该方法能够处理复杂数据结构之间的非线性关系，并在预测准确性和计算成本方面优于传统的统计模型和常规的深度学习方法。通过利用张量分解模型，提出的方法在提高预测准确性的同时，大幅度减少了数据维度从而加快了计算速度。通过模拟研究和对三个公共数据集的实际应用，验证了提出方法的实证性能，数值结果表明，与基准方法相比，提出的算法在预测准确性方面取得了显著的增加，并且计算时间显著减少。