# Noisy Tensor Completion via Low-rank Tensor Ring

arXiv: https://arxiv.org/abs/2203.08857 · HTML: https://mlsanity.com/paper/2203.08857/

## Key facts

- Cited by 13 papers in the map (data through 14 Oct 2026).
- Builds on 16 of its 77 references that are in the map.
- First posted on arXiv 14 Mar 2022.

## Builds on (most cited first, 16 of 16)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Real-world Noisy Image Denoising: A New Benchmark](https://mlsanity.com/paper/1804.02603/) (2018)
- [Bayesian CP Factorization of Incomplete Tensors with Automatic Rank Determination](https://mlsanity.com/paper/1401.6497/) (2014)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Hyperspectral Image Restoration via Total Variation Regularized Low-rank Tensor Decomposition](https://mlsanity.com/paper/1707.02477/) (2017)
- [Efficient Low Rank Tensor Ring Completion](https://mlsanity.com/paper/1707.08184/) (2017)
- [Square Deal: Lower Bounds and Improved Relaxations for Tensor Recovery](https://mlsanity.com/paper/1307.5870/) (2013)
- [On Tensor Completion via Nuclear Norm Minimization](https://mlsanity.com/paper/1405.1773/) (2014)
- [On Unifying Multi-View Self-Representations for Clustering by Tensor Multi-Rank Minimization](https://mlsanity.com/paper/1610.07126/) (2016)
- [Linked Component Analysis from Matrices to High Order Tensors: Applications to Biomedical Data](https://mlsanity.com/paper/1508.07416/) (2015)
- [High-dimension Tensor Completion via Gradient-based Optimization Under Tensor-train Format](https://mlsanity.com/paper/1804.01983/) (2018)
- [Tensor-Ring Nuclear Norm Minimization and Application for Visual Data Completion](https://mlsanity.com/paper/1903.08888/) (2019)
- [Convex Tensor Decomposition via Structured Schatten Norm Regularization](https://mlsanity.com/paper/1303.6370/) (2013)
- [Provable Tensor Ring Completion](https://mlsanity.com/paper/1903.03315/) (2019)
- [Efficient Tensor Robust PCA under Hybrid Model of Tucker and Tensor Train](https://mlsanity.com/paper/2112.10771/) (2021)
- [Simultaneous Tensor Completion and Denoising by Noise Inequality Constrained Convex Optimization](https://mlsanity.com/paper/1801.03299/) (2018)

## Built on by (most cited first, 13 of 13)

- [Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation](https://mlsanity.com/paper/2305.11495/) (2023)
- [A Generalized Tensor Formulation for Hyperspectral Image Super-Resolution Under General Spatial Blurring](https://mlsanity.com/paper/2409.18731/) (2024)
- [LoRA-PT: Low-Rank Adapting UNETR for Hippocampus Segmentation Using Principal Tensor Singular Values and Vectors](https://mlsanity.com/paper/2407.11292/) (2024)
- [A generalizable framework for low-rank tensor completion with numerical priors](https://mlsanity.com/paper/2302.05881/) (2023)
- [Tensor Ring Optimized Quantum-Enhanced Tensor Neural Networks](https://mlsanity.com/paper/2310.01515/) (2023)
- [Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration](https://mlsanity.com/paper/2407.08509/) (2024)
- [Statistical Inference in Tensor Completion: Optimal Uncertainty Quantification and Statistical-to-Computational Gaps](https://mlsanity.com/paper/2410.11225/) (2024)
- [Low-Rank Tensor Learning by Generalized Nonconvex Regularization](https://mlsanity.com/paper/2410.18402/) (2024)
- [Reparameterized Tensor Ring Functional Decomposition for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2603.01034/) (2026)
- [Quaternion tensor left ring decomposition and application for color image inpainting](https://mlsanity.com/paper/2307.10620/) (2023)
- [An Interdisciplinary and Cross-Task Review on Missing Data Imputation](https://mlsanity.com/paper/2511.01196/) (2025)
- [Exponential-Family Tensor Completion via Nonconvex Dual Total-Variation Regularization](https://mlsanity.com/paper/2606.30958/) (2026)
- [Bayesian Tensor Autoencoder with Physics-informed Predictive Prior for Multi-dimensional Time Series Anomaly Detection](https://mlsanity.com/paper/2609.31157/) (2026)

## Abstract (arXiv)

Tensor completion is a fundamental tool for incomplete data analysis, where the goal is to predict missing entries from partial observations. However, existing methods often make the explicit or implicit assumption that the observed entries are noise-free to provide a theoretical guarantee of exact recovery of missing entries, which is quite restrictive in practice. To remedy such drawbacks, this paper proposes a novel noisy tensor completion model, which complements the incompetence of existing works in handling the degeneration of high-order and noisy observations. Specifically, the tensor ring nuclear norm (TRNN) and least-squares estimator are adopted to regularize the underlying tensor and the observed entries, respectively. In addition, a non-asymptotic upper bound of estimation error is provided to depict the statistical performance of the proposed estimator. Two efficient algorithms are developed to solve the optimization problem with convergence guarantee, one of which is specially tailored to handle large-scale tensors by replacing the minimization of TRNN of the original tensor equivalently with that of a much smaller one in a heterogeneous tensor decomposition framework. Experimental results on both synthetic and real-world data demonstrate the effectiveness and efficiency of the proposed model in recovering noisy incomplete tensor data compared with state-of-the-art tensor completion models.