# High-Dimensional Low-Rank Tensor Autoregressive Time Series Modeling

arXiv: https://arxiv.org/abs/2101.04276 · HTML: https://mlsanity.com/paper/2101.04276/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2021.

## Built on by (most cited first, 5 of 5)

- [Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory](https://mlsanity.com/paper/2501.15388/) (2025)
- [Tensor Methods in High Dimensional Data Analysis: Opportunities and Challenges](https://mlsanity.com/paper/2405.18412/) (2024)
- [Orientation-Aware Sparse Tensor PCA for Efficient Unsupervised Feature Selection](https://mlsanity.com/paper/2407.16985/) (2024)
- [Sparsity-Induced Global Matrix Autoregressive Model with Auxiliary Network Data](https://mlsanity.com/paper/2503.08579/) (2025)
- [Structured Adaptive Tensor Prediction for Streaming Data](https://mlsanity.com/paper/2606.10085/) (2026)

## Abstract (arXiv)

Modern technological advances have enabled an unprecedented amount of structured data with complex temporal dependence, urging the need for new methods to efficiently model and forecast high-dimensional tensor-valued time series. This paper provides a new modeling framework to accomplish this task via autoregression (AR). By considering a low-rank Tucker decomposition for the transition tensor, the proposed tensor AR can flexibly capture the underlying low-dimensional tensor dynamics, providing both substantial dimension reduction and meaningful multi-dimensional dynamic factor interpretations. For this model, we first study several nuclear-norm-regularized estimation methods and derive their non-asymptotic properties under the approximate low-rank setting. In particular, by leveraging the special balanced structure of the transition tensor, a novel convex regularization approach based on the sum of nuclear norms of square matricizations is proposed to efficiently encourage low-rankness of the coefficient tensor. To further improve the estimation efficiency under exact low-rankness, a non-convex estimator is proposed with a gradient descent algorithm, and its computational and statistical convergence guarantees are established. Simulation studies and an empirical analysis of tensor-valued time series data from multi-category import-export networks demonstrate the advantages of the proposed approach.