# Low-rank tensor completion: a Riemannian manifold preconditioning approach

arXiv: https://arxiv.org/abs/1605.08257 · HTML: https://mlsanity.com/paper/1605.08257/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 26 May 2016.

## Built on by (most cited first, 19 of 19)

- [Scaling and Scalability: Provable Nonconvex Low-Rank Tensor Estimation from Incomplete Measurements](https://mlsanity.com/paper/2104.14526/) (2021)
- [Tensor-on-Tensor Regression: Riemannian Optimization, Over-parameterization, Statistical-computational Gap, and Their Interplay](https://mlsanity.com/paper/2206.08756/) (2022)
- [Low-rank Tensor Estimation via Riemannian Gauss-Newton: Statistical Optimality and Second-Order Convergence](https://mlsanity.com/paper/2104.12031/) (2021)
- [Subquadratic Kronecker Regression with Applications to Tensor Decomposition](https://mlsanity.com/paper/2209.04876/) (2022)
- [Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization](https://mlsanity.com/paper/2310.06159/) (2023)
- [Fast Low-Rank Tensor Decomposition by Ridge Leverage Score Sampling](https://mlsanity.com/paper/2107.10654/) (2021)
- [New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition](https://mlsanity.com/paper/2101.11108/) (2021)
- [Patch Tracking-based Streaming Tensor Ring Completion for Visual Data Recovery](https://mlsanity.com/paper/2105.14620/) (2021)
- [Streamlining in the Riemannian Realm: Efficient Riemannian Optimization with Loopless Variance Reduction](https://mlsanity.com/paper/2403.06677/) (2024)
- [Faster Convergence of Riemannian Stochastic Gradient Descent with Increasing Batch Size](https://mlsanity.com/paper/2501.18164/) (2025)
- [Low-rank Tensor Learning with Nonconvex Overlapped Nuclear Norm Regularization](https://mlsanity.com/paper/2205.03059/) (2022)
- [Partial Least Square Regression via Three-factor SVD-type Manifold Optimization for EEG Decoding](https://mlsanity.com/paper/2208.04324/) (2022)
- [Nonnegative Low-Rank Tensor Completion via Dual Formulation with Applications to Image and Video Completion](https://mlsanity.com/paper/2305.07976/) (2023)
- [Riemannian Manifold Optimization for Discriminant Subspace Learning](https://mlsanity.com/paper/2101.08032/) (2021)
- [Tensor Full Feature Measure and Its Nonconvex Relaxation Applications to Tensor Recovery](https://mlsanity.com/paper/2109.12257/) (2021)
- [Structured Low-Rank Tensor Learning](https://mlsanity.com/paper/2305.07967/) (2023)
- [Tensor Completion Leveraging Graph Information: A Dynamic Regularization Approach with Statistical Guarantees](https://mlsanity.com/paper/2310.02543/) (2023)
- [Higher Order Reduced Rank Regression](https://mlsanity.com/paper/2503.06528/) (2025)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)

## Abstract (arXiv)

We propose a novel Riemannian manifold preconditioning approach for the tensor completion problem with rank constraint. A novel Riemannian metric or inner product is proposed that exploits the least-squares structure of the cost function and takes into account the structured symmetry that exists in Tucker decomposition. The specific metric allows to use the versatile framework of Riemannian optimization on quotient manifolds to develop preconditioned nonlinear conjugate gradient and stochastic gradient descent algorithms for batch and online setups, respectively. Concrete matrix representations of various optimization-related ingredients are listed. Numerical comparisons suggest that our proposed algorithms robustly outperform state-of-the-art algorithms across different synthetic and real-world datasets.