# Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent

arXiv: https://arxiv.org/abs/2501.13385 · HTML: https://mlsanity.com/paper/2501.13385/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 19 of its 42 references that are in the map.
- First posted on arXiv 23 Jan 2025.

## Builds on (most cited first, 19 of 19)

- [Guaranteed Matrix Completion via Non-convex Factorization](https://mlsanity.com/paper/1411.8003/) (2014)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Inference and Uncertainty Quantification for Noisy Matrix Completion](https://mlsanity.com/paper/1906.04159/) (2019)
- [Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization](https://mlsanity.com/paper/2103.08895/) (2021)
- [Guarantees of Riemannian Optimization for Low Rank Matrix Completion](https://mlsanity.com/paper/1603.06610/) (2016)
- [Statistical Inferences of Linear Forms for Noisy Matrix Completion](https://mlsanity.com/paper/1909.00116/) (2019)
- [Structured Matrix Completion with Applications to Genomic Data Integration](https://mlsanity.com/paper/1504.01823/) (2015)
- [Provable Tensor-Train Format Tensor Completion by Riemannian Optimization](https://mlsanity.com/paper/2108.12163/) (2021)
- [Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trains](https://mlsanity.com/paper/1804.06128/) (2018)
- [High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format](https://mlsanity.com/paper/1804.01983/) (2018)
- [Optimal High-order Tensor SVD via Tensor-Train Orthogonal Iteration](https://mlsanity.com/paper/2010.02482/) (2020)
- [Quantum State Tomography for Matrix Product Density Operators](https://mlsanity.com/paper/2306.09432/) (2023)
- [Tensor Completion by Alternating Minimization under the Tensor Train (TT) Model](https://mlsanity.com/paper/1609.05587/) (2016)
- [Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion](https://mlsanity.com/paper/2108.07899/) (2021)
- [A Preconditioned Riemannian Gradient Descent Algorithm for Low-Rank Matrix Recovery](https://mlsanity.com/paper/2305.02543/) (2023)
- [On Tensor Train Rank Minimization: Statistical Efficiency and Scalable Algorithm](https://mlsanity.com/paper/1708.00132/) (2017)
- [New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition](https://mlsanity.com/paper/2101.11108/) (2021)
- [Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metric](https://mlsanity.com/paper/2209.04786/) (2022)
- [A Single-Mode Quasi Riemannian Gradient Descent Algorithm for Low-Rank Tensor Recovery](https://mlsanity.com/paper/2401.15925/) (2024)

## Abstract (arXiv)

Low-rank tensor completion aims to recover a tensor from partially observed entries, and it is widely applicable in fields such as quantum computing and image processing. Due to the significant advantages of the tensor train (TT) format in handling structured high-order tensors, this paper investigates the low-rank tensor completion problem based on the TT-format. We proposed a preconditioned Riemannian gradient descent algorithm (PRGD) to solve low TT-rank tensor completion and establish its linear convergence. Experimental results on both simulated and real datasets demonstrate the effectiveness of the PRGD algorithm. On the simulated dataset, the PRGD algorithm reduced the computation time by two orders of magnitude compared to existing classical algorithms. In practical applications such as hyperspectral image completion and quantum state tomography, the PRGD algorithm significantly reduced the number of iterations, thereby substantially reducing the computational time.