# Implicit Regularization and Entrywise Convergence of Riemannian Optimization for Low Tucker-Rank Tensor Completion

arXiv: https://arxiv.org/abs/2108.07899 · HTML: https://mlsanity.com/paper/2108.07899/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 17 Aug 2021.

## Built on by (most cited first, 9 of 9)

- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [Provably Accelerating Ill-Conditioned Low-rank Estimation via Scaled Gradient Descent, Even with Overparameterization](https://mlsanity.com/paper/2310.06159/) (2023)
- [Online Tensor Learning: Computational and Statistical Trade-offs, Adaptivity and Optimal Regret](https://mlsanity.com/paper/2306.03372/) (2023)
- [Leave-One-Out Analysis for Nonconvex Robust Matrix Completion with General Thresholding Functions](https://mlsanity.com/paper/2407.19446/) (2024)
- [Tensor Completion Leveraging Graph Information: A Dynamic Regularization Approach with Statistical Guarantees](https://mlsanity.com/paper/2310.02543/) (2023)
- [Fast and Provable Tensor-Train Format Tensor Completion via Precondtioned Riemannian Gradient Descent](https://mlsanity.com/paper/2501.13385/) (2025)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)
- [Tensor Completion using Subspace Information](https://mlsanity.com/paper/2609.24501/) (2026)
- [Guaranteed Low-Rank Tensor Recovery from Modewise Measurements via Normalized Block-Weighted Riemannian Gradient Descent](https://mlsanity.com/paper/2609.24679/) (2026)

## Abstract (arXiv)

This paper is concerned with the low Tucker-rank tensor completion problem, which is about reconstructing a tensor  T ∈ℝn× n × n of low multilinear rank from partially observed entries. Riemannian optimization algorithms are a class of efficient methods for this problem, but the theoretical convergence analysis is still lacking. In this manuscript, we establish the entrywise convergence of the vanilla Riemannian gradient method for low Tucker-rank tensor completion under the nearly optimal sampling complexity O(n3/2). Meanwhile, the implicit regularization phenomenon of the algorithm has also been revealed. As far as we know, this is the first work that has shown the entrywise convergence and implicit regularization property of a non-convex method for low Tucker-rank tensor completion. The analysis relies on the leave-one-out technique, and some of the technical results developed in the paper might be of broader interest in investigating the properties of other non-convex methods for this problem.