# Tensor Completion via Tensor Train Based Low-Rank Quotient Geometry under a Preconditioned Metric

arXiv: https://arxiv.org/abs/2209.04786 · HTML: https://mlsanity.com/paper/2209.04786/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 11 Sep 2022.

## Built on by (most cited first, 3 of 3)

- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Fourier Low-rank and Sparse Tensor for Efficient Tensor Completion](https://mlsanity.com/paper/2505.11261/) (2025)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)

## Abstract (arXiv)

This paper investigates the low-rank tensor completion problem, which is about recovering a tensor from partially observed entries. We consider this problem in the tensor train format and extend the preconditioned metric from the matrix case to the tensor case. The first-order and second-order quotient geometry of the manifold of fixed tensor train rank tensors under this metric is studied in detail. Algorithms, including Riemannian gradient descent, Riemannian conjugate gradient, and Riemannian Gauss-Newton, have been proposed for the tensor completion problem based on the quotient geometry. It has also been shown that the Riemannian Gauss-Newton method on the quotient geometry is equivalent to the Riemannian Gauss-Newton method on the embedded geometry with a specific retraction. Empirical evaluations on random instances as well as on function-related tensors show that the proposed algorithms are competitive with other existing algorithms in terms of recovery ability, convergence performance, and reconstruction quality.