# High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format

arXiv: https://arxiv.org/abs/1804.01983 · HTML: https://mlsanity.com/paper/1804.01983/

## Key facts

- Cited by 12 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 5 Apr 2018.

## Built on by (most cited first, 12 of 12)

- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Noisy Tensor Completion via Low-rank Tensor Ring](https://mlsanity.com/paper/2203.08857/) (2022)
- [Permutation Search of Tensor Network Structures via Local Sampling](https://mlsanity.com/paper/2206.06597/) (2022)
- [SVDinsTN: A Tensor Network Paradigm for Efficient Structure Search from Regularized Modeling Perspective](https://mlsanity.com/paper/2305.14912/) (2023)
- [Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery](https://mlsanity.com/paper/2401.02592/) (2024)
- [More Efficient Sampling for Tensor Decomposition With Worst-Case Guarantees](https://mlsanity.com/paper/2110.07631/) (2021)
- [A Scalable Factorization Approach for High-Order Structured Tensor Recovery](https://mlsanity.com/paper/2506.16032/) (2025)
- [On Model Compression for Neural Networks: Framework, Algorithm, and Convergence Guarantee](https://mlsanity.com/paper/2303.06815/) (2023)
- [Tensor Train Completion from Fiberwise Observations Along a Single Mode](https://mlsanity.com/paper/2509.18149/) (2025)
- [Streaming data recovery via Bayesian tensor train decomposition](https://mlsanity.com/paper/2302.12148/) (2023)
- [Optical Coherence Tomography Image Enhancement via Block Hankelization and Low Rank Tensor Network Approximation](https://mlsanity.com/paper/2306.11750/) (2023)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)

## Abstract (arXiv)

Tensor train (TT) decomposition has drawn people's attention due to its powerful representation ability and performance stability in high-order tensors. In this paper, we propose a novel approach to recover the missing entries of incomplete data represented by higher-order tensors. We attempt to find the low-rank TT decomposition of the incomplete data which captures the latent features of the whole data and then reconstruct the missing entries. By applying gradient descent algorithms, tensor completion problem is efficiently solved by optimization models. We propose two TT-based algorithms: Tensor Train Weighted Optimization (TT-WOPT) and Tensor Train Stochastic Gradient Descent (TT-SGD) to optimize TT decomposition factors. In addition, a method named Visual Data Tensorization (VDT) is proposed to transform visual data into higher-order tensors, resulting in the performance improvement of our algorithms. The experiments in synthetic data and visual data show high efficiency and performance of our algorithms compared to the state-of-the-art completion algorithms, especially in high-order, high missing rate, and large-scale tensor completion situations.