# Guaranteed Tensor Recovery Fused Low-rankness and Smoothness

arXiv: https://arxiv.org/abs/2302.02155 · HTML: https://mlsanity.com/paper/2302.02155/

## Key facts

- Cited by 41 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 85 references that are in the map.
- First posted on arXiv 4 Feb 2023.

## Builds on (most cited first, 24 of 24)

- [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)
- [Exact tensor completion using t-SVD](https://mlsanity.com/paper/1502.04689/) (2015)
- [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)
- [Tensor Methods in Computer Vision and Deep Learning](https://mlsanity.com/paper/2107.03436/) (2021)
- [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)
- [Robust Low-rank Tensor Recovery: Models and Algorithms](https://mlsanity.com/paper/1311.6182/) (2013)
- [Smooth PARAFAC Decomposition for Tensor Completion](https://mlsanity.com/paper/1505.06611/) (2015)
- [Tensor-based formulation and nuclear norm regularization for multi-energy computed tomography](https://mlsanity.com/paper/1307.5348/) (2013)
- [Bayesian Robust Tensor Factorization for Incomplete Multiway Data](https://mlsanity.com/paper/1410.2386/) (2014)
- [Exact Decomposition of Joint Low Rankness and Local Smoothness Plus Sparse Matrices](https://mlsanity.com/paper/2201.12592/) (2022)
- [Low-tubal-rank Tensor Completion using Alternating Minimization](https://mlsanity.com/paper/1610.01690/) (2016)
- [Fast and Accurate Tensor Completion with Total Variation Regularized Tensor Trains](https://mlsanity.com/paper/1804.06128/) (2018)
- [Fast Noise Removal in Hyperspectral Images via Representative Coefficient Total Variation](https://mlsanity.com/paper/2211.01825/) (2022)
- [Large-Scale Low-Rank Matrix Learning with Nonconvex Regularizers](https://mlsanity.com/paper/1708.00146/) (2017)
- [A New Convex Relaxation for Tensor Completion](https://mlsanity.com/paper/1307.4653/) (2013)
- [Convex Tensor Decomposition via Structured Schatten Norm Regularization](https://mlsanity.com/paper/1303.6370/) (2013)
- [Collaborative Total Variation: A General Framework for Vectorial TV Models](https://mlsanity.com/paper/1508.01308/) (2015)
- [Tensor vs Matrix Methods: Robust Tensor Decomposition under Block Sparse Perturbations](https://mlsanity.com/paper/1510.04747/) (2015)
- [Guarantees of Total Variation Minimization for Signal Recovery](https://mlsanity.com/paper/1301.6791/) (2013)
- [Projected Robust PCA with Application to Smooth Image Recovery](https://mlsanity.com/paper/2009.05478/) (2020)

## Built on by (most cited first, 30 of 41)

- [Revisiting Nonlocal Self-Similarity from Continuous Representation](https://mlsanity.com/paper/2401.00708/) (2024)
- [Nonconvex Robust High-Order Tensor Completion Using Randomized Low-Rank Approximation](https://mlsanity.com/paper/2305.11495/) (2023)
- [Low-Tubal-Rank Tensor Recovery via Factorized Gradient Descent](https://mlsanity.com/paper/2401.11940/) (2024)
- [NeurTV: Total Variation on the Neural Domain](https://mlsanity.com/paper/2405.17241/) (2024)
- [Pan-denoising: Guided Hyperspectral Image Denoising via Weighted Represent Coefficient Total Variation](https://mlsanity.com/paper/2407.06064/) (2024)
- [Hipandas: Hyperspectral Image Joint Denoising and Super-Resolution by Image Fusion with the Panchromatic Image](https://mlsanity.com/paper/2412.04201/) (2024)
- [Cross-Frequency Implicit Neural Representation with Self-Evolving Parameters](https://mlsanity.com/paper/2504.10929/) (2025)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Guaranteed Multidimensional Time Series Prediction via Deterministic Tensor Completion Theory](https://mlsanity.com/paper/2501.15388/) (2025)
- [Hyperspectral Anomaly Detection Fused Unified Nonconvex Tensor Ring Factors Regularization](https://mlsanity.com/paper/2505.17881/) (2025)
- [It's Not the Target, It's the Background: Rethinking Infrared Small Target Detection via Deep Patch-Free Low-Rank Representations](https://mlsanity.com/paper/2506.10425/) (2025)
- [Power of ℓ1-Norm Regularized Kaczmarz Algorithms for High-Order Tensor Recovery](https://mlsanity.com/paper/2405.08275/) (2024)
- [Haar Nuclear Norms with Applications to Remote Sensing Imagery Restoration](https://mlsanity.com/paper/2407.08509/) (2024)
- [Low-Rank Tensor Recovery via Variational Schatten-p Quasi-Norm and Jacobian Regularization](https://mlsanity.com/paper/2506.22134/) (2025)
- [Beyond Motion Cues and Structural Sparsity: Revisiting Small Moving Target Detection](https://mlsanity.com/paper/2509.07654/) (2025)
- [Gaussian Splatting-based Low-Rank Tensor Representation for Multi-Dimensional Image Recovery](https://mlsanity.com/paper/2511.14270/) (2025)
- [Robust Low-Rank Tensor Completion based on M-product with Weighted Correlated Total Variation and Sparse Regularization](https://mlsanity.com/paper/2604.13525/) (2026)
- [Neural Gradient Regularizer](https://mlsanity.com/paper/2308.16612/) (2023)
- [Variational Bayesian Inference for Tensor Robust Principal Component Analysis](https://mlsanity.com/paper/2412.18717/) (2024)
- [Outlier-aware Tensor Robust Principal Component Analysis with Self-guided Data Augmentation](https://mlsanity.com/paper/2504.18323/) (2025)
- [Accelerating Large-Scale Regularized High-Order Tensor Recovery](https://mlsanity.com/paper/2506.09594/) (2025)
- [3DeepRep: 3D Deep Low-rank Tensor Representation for Hyperspectral Image Inpainting](https://mlsanity.com/paper/2506.16735/) (2025)
- [Robust Tensor Completion via Gradient Tensor Nulclear L1-L2 Norm for Traffic Data Recovery](https://mlsanity.com/paper/2506.22732/) (2025)
- [Beyond Low-rankness: Guaranteed Matrix Recovery via Modified Nuclear Norm](https://mlsanity.com/paper/2507.18327/) (2025)
- [Hyperspectral Image Recovery Constrained by Multi-Granularity Non-Local Self-Similarity Priors](https://mlsanity.com/paper/2508.01435/) (2025)
- [LRTuckerRep: Low-rank Tucker Representation Model for Multi-dimensional Data Completion](https://mlsanity.com/paper/2508.03755/) (2025)
- [Superpixel-informed Continuous Low-Rank Tensor Representation for Multi-Dimensional Data Recovery](https://mlsanity.com/paper/2508.12261/) (2025)
- [Generalized Nonnegative Structured Kruskal Tensor Regression](https://mlsanity.com/paper/2509.19900/) (2025)
- [Compact Multi-level-prior Tensor Representation for Hyperspectral Image Super-resolution](https://mlsanity.com/paper/2510.06098/) (2025)
- [Tensor Completion via Monotone Inclusion: Generalized Low-Rank Priors Meet Deep Denoisers](https://mlsanity.com/paper/2510.12425/) (2025)

## Abstract (arXiv)

The tensor data recovery task has thus attracted much research attention in recent years. Solving such an ill-posed problem generally requires to explore intrinsic prior structures underlying tensor data, and formulate them as certain forms of regularization terms for guiding a sound estimate of the restored tensor. Recent research have made significant progress by adopting two insightful tensor priors, i.e., global low-rankness (L) and local smoothness (S) across different tensor modes, which are always encoded as a sum of two separate regularization terms into the recovery models. However, unlike the primary theoretical developments on low-rank tensor recovery, these joint L+S models have no theoretical exact-recovery guarantees yet, making the methods lack reliability in real practice. To this crucial issue, in this work, we build a unique regularization term, which essentially encodes both L and S priors of a tensor simultaneously. Especially, by equipping this single regularizer into the recovery models, we can rigorously prove the exact recovery guarantees for two typical tensor recovery tasks, i.e., tensor completion (TC) and tensor robust principal component analysis (TRPCA). To the best of our knowledge, this should be the first exact-recovery results among all related L+S methods for tensor recovery. Significant recovery accuracy improvements over many other SOTA methods in several TC and TRPCA tasks with various kinds of visual tensor data are observed in extensive experiments. Typically, our method achieves a workable performance when the missing rate is extremely large, e.g., 99.5%, for the color image inpainting task, while all its peers totally fail in such challenging case.