Tensor Time Series: Factor Modeling and Deep Neural Networks

Yuefeng Han, Assistant Professor, Department of Applied and Computational Mathematics and Statistics, University of Notre Dame

Abstract: The analysis of tensors (multi-dimensional arrays) has become a vital area in modern statistics and data science, driven by advancements in scientific research and data collection. High-dimensional tensor data arise in diverse applications such as economics, genetics, microbiome studies, brain imaging, and hyperspectral imaging. These tensors are often high-dimensional and high-order, yet key information typically resides in reduced-dimensional subspaces governed by structural properties. This talk explores novel methodologies and theories for tensor time series analysis.

The presentation consists of two parts. The first part introduces a factor modeling framework for high-dimensional tensor time series, leveraging a structure similar to CP tensor decomposition. We propose a computationally efficient estimation procedure incorporating a warm-start initialization and an iterative simultaneous orthogonalization scheme. The algorithm achieves $\epsilon$-accuracy within $\log\log(1/\epsilon)$ iterations. Additionally, we establish inferential results, demonstrating consistency and asymptotic normality under relaxed assumptions. The second part integrates tensor factor models with deep neural networks. Specifically, a Tucker-type low-rank tensor structure is employed as a tensor-augmentation module in neural networks. Extensive experiments demonstrate the integration of this module into transformers and temporal neural networks for tensor time series prediction and tensor-on-tensor regression. The results highlight significant performance improvements, underscoring its potential for advancing time series forecasting.

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