# New Riemannian preconditioned algorithms for tensor completion via polyadic decomposition

arXiv: https://arxiv.org/abs/2101.11108 · HTML: https://mlsanity.com/paper/2101.11108/

## Key facts

- Cited by 4 papers in the map (data through 14 Oct 2026).
- Builds on 8 of its 51 references that are in the map.
- First posted on arXiv 26 Jan 2021.

## Builds on (most cited first, 8 of 8)

- [Manopt, a Matlab toolbox for optimization on manifolds](https://mlsanity.com/paper/1308.5200/) (2013)
- [Global rates of convergence for nonconvex optimization on manifolds](https://mlsanity.com/paper/1605.08101/) (2016)
- [Smooth PARAFAC Decomposition for Tensor Completion](https://mlsanity.com/paper/1505.06611/) (2015)
- [Provable Tensor Factorization with Missing Data](https://mlsanity.com/paper/1406.2784/) (2014)
- [Riemannian preconditioning](https://mlsanity.com/paper/1405.6055/) (2014)
- [Low-rank tensor completion: a Riemannian manifold preconditioning approach](https://mlsanity.com/paper/1605.08257/) (2016)
- [Convergence results for projected line-search methods on varieties of low-rank matrices via Lojasiewicz inequality](https://mlsanity.com/paper/1402.5284/) (2014)
- [Alternating minimization algorithms for graph regularized tensor completion](https://mlsanity.com/paper/2008.12876/) (2020)

## Built on by (most cited first, 4 of 4)

- [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)
- [Manifold constrained steepest descent for smooth and closed-set optimization](https://mlsanity.com/paper/2601.21487/) (2026)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)

## Abstract (arXiv)

We propose new Riemannian preconditioned algorithms for low-rank tensor completion via the polyadic decomposition of a tensor. These algorithms exploit a non-Euclidean metric on the product space of the factor matrices of the low-rank tensor in the polyadic decomposition form. This new metric is designed using an approximation of the diagonal blocks of the Hessian of the tensor completion cost function, thus has a preconditioning effect on these algorithms. We prove that the proposed Riemannian gradient descent algorithm globally converges to a stationary point of the tensor completion problem, with convergence rate estimates using the Lojasiewicz property. Numerical results on synthetic and real-world data suggest that the proposed algorithms are more efficient in memory and time compared to state-of-the-art algorithms. Moreover, the proposed algorithms display a greater tolerance for overestimated rank parameters in terms of the tensor recovery performance, thus enable a flexible choice of the rank parameter.