# Tensor Completion by Alternating Minimization under the Tensor Train (TT) Model

arXiv: https://arxiv.org/abs/1609.05587 · HTML: https://mlsanity.com/paper/1609.05587/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 19 Sep 2016.

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

- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery](https://mlsanity.com/paper/2401.02592/) (2024)
- [TenIPS: Inverse Propensity Sampling for Tensor Completion](https://mlsanity.com/paper/2101.00323/) (2021)
- [A Scalable Factorization Approach for High-Order Structured Tensor Recovery](https://mlsanity.com/paper/2506.16032/) (2025)
- [Lower and Upper Bounds on the VC-Dimension of Tensor Network Models](https://mlsanity.com/paper/2106.11827/) (2021)
- [Tensor Network for Supervised Learning at Finite Temperature](https://mlsanity.com/paper/2104.05439/) (2021)
- [Streaming data recovery via Bayesian tensor train decomposition](https://mlsanity.com/paper/2302.12148/) (2023)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)
- [Geological Field Restoration through the Lens of Image Inpainting](https://mlsanity.com/paper/2506.04869/) (2025)

## Abstract (arXiv)

Using the matrix product state (MPS) representation of tensor train decompositions, in this paper we propose a tensor completion algorithm which alternates over the matrices (tensors) in the MPS representation. This development is motivated in part by the success of matrix completion algorithms which alternate over the (low-rank) factors. We comment on the computational complexity of the proposed algorithm and numerically compare it with existing methods employing low rank tensor train approximation for data completion as well as several other recently proposed methods. We show that our method is superior to existing ones for a variety of real settings.