# Fully-Connected Tensor Network Decomposition for Robust Tensor Completion Problem

arXiv: https://arxiv.org/abs/2110.08754 · HTML: https://mlsanity.com/paper/2110.08754/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 9 of its 52 references that are in the map.
- First posted on arXiv 17 Oct 2021.

## Builds on (most cited first, 9 of 9)

- [Tensor Ring Decomposition](https://mlsanity.com/paper/1606.05535/) (2016)
- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Efficient tensor completion for color image and video recovery: Low-rank tensor train](https://mlsanity.com/paper/1606.01500/) (2016)
- [Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography](https://mlsanity.com/paper/1307.5348/) (2013)
- [Guarantees of Riemannian Optimization for Low Rank Matrix Completion](https://mlsanity.com/paper/1603.06610/) (2016)
- [Tensor-Ring Nuclear Norm Minimization and Application for Visual Data Completion](https://mlsanity.com/paper/1903.08888/) (2019)
- [Tensor train rank minimization with nonlocal self-similarity for tensor completion](https://mlsanity.com/paper/2004.14273/) (2020)
- [Bilinear Constraint based ADMM for Mixed Poisson-Gaussian Noise Removal](https://mlsanity.com/paper/1910.08206/) (2019)
- [Exact Tensor Completion from Sparsely Corrupted Observations via Convex Optimization](https://mlsanity.com/paper/1708.00601/) (2017)

## Built on by (most cited first, 5 of 5)

- [A high-order tensor completion algorithm based on Fully-Connected Tensor Network weighted optimization](https://mlsanity.com/paper/2204.01732/) (2022)
- [Latent Matrices for Tensor Network Decomposition and to Tensor Completion](https://mlsanity.com/paper/2210.03392/) (2022)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [Tensor Completion via Monotone Inclusion: Generalized Low-Rank Priors Meet Deep Denoisers](https://mlsanity.com/paper/2510.12425/) (2025)
- [Multi-Dimensional Visual Data Recovery: Scale-Aware Tensor Modeling and Accelerated Randomized Computation](https://mlsanity.com/paper/2602.12982/) (2026)

## Abstract (arXiv)

The robust tensor completion (RTC) problem, which aims to reconstruct a low-rank tensor from partially observed tensor contaminated by a sparse tensor, has received increasing attention. In this paper, by leveraging the superior expression of the fully-connected tensor network (FCTN) decomposition, we propose a FCTN-based robust convex optimization model (RC-FCTN) for the RTC problem. Then, we rigorously establish the exact recovery guarantee for the RC-FCTN. For solving the constrained optimization model RC-FCTN, we develop an alternating direction method of multipliers (ADMM)-based algorithm, which enjoys the global convergence guarantee. Moreover, we suggest a FCTN-based robust nonconvex optimization model (RNC-FCTN) for the RTC problem. A proximal alternating minimization (PAM)-based algorithm is developed to solve the proposed RNC-FCTN. Meanwhile, we theoretically derive the convergence of the PAM-based algorithm. Comprehensive numerical experiments in several applications, such as video completion and video background subtraction, demonstrate that proposed methods are superior to several state-of-the-art methods.