# Robust Tensor CUR Decompositions: Rapid Low-Tucker-Rank Tensor Recovery with Sparse Corruption

arXiv: https://arxiv.org/abs/2305.04080 · HTML: https://mlsanity.com/paper/2305.04080/

## Key facts

- Cited by 11 papers in the map (data through 14 Oct 2026).
- Builds on 21 of its 63 references that are in the map.
- First posted on arXiv 6 May 2023.

## Builds on (most cited first, 21 of 21)

- [Tensor Robust Principal Component Analysis with A New Tensor Nuclear Norm](https://mlsanity.com/paper/1804.03728/) (2018)
- [Novel methods for multilinear data completion and de-noising based on tensor-SVD](https://mlsanity.com/paper/1407.1785/) (2014)
- [Introduction to Tensor Decompositions and their Applications in Machine Learning](https://mlsanity.com/paper/1711.10781/) (2017)
- [Non-convex Robust PCA](https://mlsanity.com/paper/1410.7660/) (2014)
- [Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization](https://mlsanity.com/paper/1708.04181/) (2017)
- [Learned Robust PCA: A Scalable Deep Unfolding Approach for High-Dimensional Outlier Detection](https://mlsanity.com/paper/2110.05649/) (2021)
- [Tucker Tensor Regression and Neuroimaging Analysis](https://mlsanity.com/paper/1304.5637/) (2013)
- [Accelerated Alternating Projections for Robust Principal Component Analysis](https://mlsanity.com/paper/1711.05519/) (2017)
- [Generalized Low-rank plus Sparse Tensor Estimation by Fast Riemannian Optimization](https://mlsanity.com/paper/2103.08895/) (2021)
- [Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entries](https://mlsanity.com/paper/1711.04934/) (2017)
- [Coauthorship and Citation Networks for Statisticians](https://mlsanity.com/paper/1410.2840/) (2014)
- [Robust CUR Decomposition: Theory and Imaging Applications](https://mlsanity.com/paper/2101.05231/) (2021)
- [Rapid Robust Principal Component Analysis: CUR Accelerated Inexact Low Rank Estimation](https://mlsanity.com/paper/2010.07422/) (2020)
- [Mode-wise Tensor Decompositions: Multi-dimensional Generalizations of CUR Decompositions](https://mlsanity.com/paper/2103.11037/) (2021)
- [Structured Gradient Descent for Fast Robust Low-Rank Hankel Matrix Completion](https://mlsanity.com/paper/2204.03316/) (2022)
- [Fast and Provable Tensor Robust Principal Component Analysis via Scaled Gradient Descent](https://mlsanity.com/paper/2206.09109/) (2022)
- [Characterizing Spatiotemporal Transcriptome of Human Brain via Low Rank Tensor Decomposition](https://mlsanity.com/paper/1702.07449/) (2017)
- [Fast Robust Tensor Principal Component Analysis via Fiber CUR Decomposition](https://mlsanity.com/paper/2108.10448/) (2021)
- [Riemannian CUR Decompositions for Robust Principal Component Analysis](https://mlsanity.com/paper/2206.09042/) (2022)
- [Static and Dynamic Robust PCA and Matrix Completion: A Review](https://mlsanity.com/paper/1803.00651/) (2018)
- [Robust Tensor Recovery with Fiber Outliers for Traffic Events](https://mlsanity.com/paper/1908.10198/) (2019)

## Built on by (most cited first, 11 of 11)

- [On the Robustness of Cross-Concentrated Sampling for Matrix Completion](https://mlsanity.com/paper/2401.15566/) (2024)
- [Computational and Statistical Guarantees for Tensor-on-Tensor Regression with Tensor Train Decomposition](https://mlsanity.com/paper/2406.06002/) (2024)
- [Accelerating Ill-conditioned Hankel Matrix Recovery via Structured Newton-like Descent](https://mlsanity.com/paper/2406.07409/) (2024)
- [S3Attention: Improving Long Sequence Attention with Smoothed Skeleton Sketching](https://mlsanity.com/paper/2408.08567/) (2024)
- [Robust Low-rank Tensor Train Recovery](https://mlsanity.com/paper/2410.15224/) (2024)
- [Learnable Scaled Gradient Descent for Guaranteed Robust Tensor PCA](https://mlsanity.com/paper/2501.04565/) (2025)
- [Variational Bayesian Inference for Tensor Robust Principal Component Analysis](https://mlsanity.com/paper/2412.18717/) (2024)
- [Robust Anomaly Detection via Tensor Pseudoskeleton Decomposition](https://mlsanity.com/paper/2502.09926/) (2025)
- [Property Inheritance for Subtensors in Tensor Train Decompositions](https://mlsanity.com/paper/2504.11396/) (2025)
- [Outlier-aware Tensor Robust Principal Component Analysis with Self-guided Data Augmentation](https://mlsanity.com/paper/2504.18323/) (2025)
- [Robust Low-Tubal-Rank Tensor Completion under Cross-Concentrated Sampling](https://mlsanity.com/paper/2608.03928/) (2026)

## Abstract (arXiv)

We study the tensor robust principal component analysis (TRPCA) problem, a tensorial extension of matrix robust principal component analysis (RPCA), that aims to split the given tensor into an underlying low-rank component and a sparse outlier component. This work proposes a fast algorithm, called Robust Tensor CUR Decompositions (RTCUR), for large-scale non-convex TRPCA problems under the Tucker rank setting. RTCUR is developed within a framework of alternating projections that projects between the set of low-rank tensors and the set of sparse tensors. We utilize the recently developed tensor CUR decomposition to substantially reduce the computational complexity in each projection. In addition, we develop four variants of RTCUR for different application settings. We demonstrate the effectiveness and computational advantages of RTCUR against state-of-the-art methods on both synthetic and real-world datasets.