# Tensor train completion: local recovery guarantees via Riemannian optimization

arXiv: https://arxiv.org/abs/2110.03975 · HTML: https://mlsanity.com/paper/2110.03975/

## Key facts

- Cited by 6 papers in the map (data through 14 Oct 2026).
- Builds on 10 of its 62 references that are in the map.
- First posted on arXiv 8 Oct 2021.

## Builds on (most cited first, 10 of 10)

- [Gradient Descent with Random Initialization: Fast Global Convergence for Nonconvex Phase Retrieval](https://mlsanity.com/paper/1803.07726/) (2018)
- [Implicit Regularization in Nonconvex Statistical Estimation: Gradient Descent Converges Linearly for Phase Retrieval, Matrix Completion, and Blind Deconvolution](https://mlsanity.com/paper/1711.10467/) (2017)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery](https://mlsanity.com/paper/1307.5870/) (2013)
- [Provable Inductive Matrix Completion](https://mlsanity.com/paper/1306.0626/) (2013)
- [On Tensor Completion via Nuclear Norm Minimization](https://mlsanity.com/paper/1405.1773/) (2014)
- [Guarantees of Riemannian Optimization for Low Rank Matrix Completion](https://mlsanity.com/paper/1603.06610/) (2016)
- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality](https://mlsanity.com/paper/1402.5284/) (2014)
- [Incoherent Tensor Norms and Their Applications in Higher Order Tensor Completion](https://mlsanity.com/paper/1606.03504/) (2016)

## Built on by (most cited first, 6 of 6)

- [Guaranteed Nonconvex Factorization Approach for Tensor Train Recovery](https://mlsanity.com/paper/2401.02592/) (2024)
- [Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition](https://mlsanity.com/paper/2406.06002/) (2024)
- [A Scalable Factorization Approach for High-Order Structured Tensor Recovery](https://mlsanity.com/paper/2506.16032/) (2025)
- [Robust Low-rank Tensor Train Recovery](https://mlsanity.com/paper/2410.15224/) (2024)
- [Variational Bayesian inference for CP tensor completion with side information](https://mlsanity.com/paper/2206.12486/) (2022)
- [Efficient Tensor Completion Algorithms for Highly Oscillatory Operators](https://mlsanity.com/paper/2510.17734/) (2025)

## Abstract (arXiv)

In this work, we estimate the number of randomly selected elements of a tensor that with high probability guarantees local convergence of Riemannian gradient descent for tensor train completion. We derive a new bound for the orthogonal projections onto the tangent spaces based on the harmonic mean of the unfoldings' singular values and introduce a notion of core coherence for tensor trains. We also extend the results to tensor train completion with auxiliary subspace information and obtain the corresponding local convergence guarantees.