# Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration

arXiv: https://arxiv.org/abs/2010.02482 · HTML: https://mlsanity.com/paper/2010.02482/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 6 Oct 2020.

## Built on by (most cited first, 11 of 11)

- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [TT-NF: Tensor Train Neural Fields](https://mlsanity.com/paper/2209.15529/) (2022)
- [Deflated HeteroPCA: Overcoming the curse of ill-conditioning in heteroskedastic PCA](https://mlsanity.com/paper/2303.06198/) (2023)
- [To Fold or Not to Fold: Graph Regularized Tensor Train for Visual Data Completion](https://mlsanity.com/paper/2306.11123/) (2023)
- [Multivariate Analysis for Multiple Network Data via Semi-Symmetric Tensor PCA](https://mlsanity.com/paper/2202.04719/) (2022)
- [Mode-wise Principal Subspace Pursuit and Matrix Spiked Covariance Model](https://mlsanity.com/paper/2307.00575/) (2023)
- [Tensor Decomposition with Unaligned Observations](https://mlsanity.com/paper/2410.14046/) (2024)
- [Cluster Quilting: Spectral Clustering for Patchwork Learning](https://mlsanity.com/paper/2406.13833/) (2024)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)
- [A Robust and Non-Iterative Tensor Decomposition Method with Automatic Thresholding](https://mlsanity.com/paper/2505.06203/) (2025)

## Abstract (arXiv)

This paper studies a general framework for high-order tensor SVD. We propose a new computationally efficient algorithm, tensor-train orthogonal iteration (TTOI), that aims to estimate the low tensor-train rank structure from the noisy high-order tensor observation. The proposed TTOI consists of initialization via TT-SVD (Oseledets, 2011) and new iterative backward/forward updates. We develop the general upper bound on estimation error for TTOI with the support of several new representation lemmas on tensor matricizations. By developing a matching information-theoretic lower bound, we also prove that TTOI achieves the minimax optimality under the spiked tensor model. The merits of the proposed TTOI are illustrated through applications to estimation and dimension reduction of high-order Markov processes, numerical studies, and a real data example on New York City taxi travel records. The software of the proposed algorithm is available online6.