# Tensor Factor Model Estimation by Iterative Projection

arXiv: https://arxiv.org/abs/2006.02611 · HTML: https://mlsanity.com/paper/2006.02611/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 4 Jun 2020.

## Built on by (most cited first, 9 of 9)

- [Tensor Principal Component Analysis in High Dimensional CP Models](https://mlsanity.com/paper/2108.04428/) (2021)
- [Factor Augmented Tensor-on-Tensor Neural Networks](https://mlsanity.com/paper/2405.19610/) (2024)
- [Multi-dimensional domain generalization with low-rank structures](https://mlsanity.com/paper/2309.09555/) (2023)
- [Covariate-Adjusted Deep Causal Learning for Heterogeneous Panel Data Models](https://mlsanity.com/paper/2505.20536/) (2025)
- [Tucker Diffusion Model for High-dimensional Tensor Generation](https://mlsanity.com/paper/2604.00481/) (2026)
- [TEAFormers: TEnsor-Augmented Transformers for Multi-Dimensional Time Series Forecasting](https://mlsanity.com/paper/2410.20439/) (2024)
- [Supervised Dynamic Dimension Reduction with Deep Neural Network](https://mlsanity.com/paper/2508.03546/) (2025)
- [Modeling High-Dimensional Dependent Data in the Presence of Many Explanatory Variables and Weak Signals](https://mlsanity.com/paper/2412.04736/) (2024)
- [Factor Augmented High-Dimensional SGD](https://mlsanity.com/paper/2605.19291/) (2026)

## Abstract (arXiv)

Tensor time series, which is a time series consisting of tensorial observations, has become ubiquitous. It typically exhibits high dimensionality. One approach for dimension reduction is to use a factor model structure, in a form similar to Tucker tensor decomposition, except that the time dimension is treated as a dynamic process with a time dependent structure. In this paper we introduce two approaches to estimate such a tensor factor model by using iterative orthogonal projections of the original tensor time series. These approaches extend the existing estimation procedures and improve the estimation accuracy and convergence rate significantly as proven in our theoretical investigation. Our algorithms are similar to the higher order orthogonal projection method for tensor decomposition, but with significant differences due to the need to unfold tensors in the iterations and the use of autocorrelation. Consequently, our analysis is significantly different from the existing ones. Computational and statistical lower bounds are derived to prove the optimality of the sample size requirement and convergence rate for the proposed methods. Simulation study is conducted to further illustrate the statistical properties of these estimators.