Tensor Robust Principal Component Analysis: Exact Recovery of Corrupted Low-Rank Tensors via Convex Optimization

By Canyi Lu, Jiashi Feng, Yudong Chen, Wei David Liu, Zhouchen Lin, Shuicheng Yan. Computer Vision and Pattern Recognition, 2016

528 citations recorded by the source corpus.

This paper studies the Tensor Robust Principal Component (TRPCA) problem which extends the known Robust PCA [4] to the tensor case. Our model is based on a new tensor Singular Value Decomposition (t-SVD) [14] and its induced tensor tubal rank and tensor nuclear norm. Consider that we have a 3-way tensor X ε R^n1×n2×n3 such that X = L₀ + S₀, where L₀ has low tubal rank and _{S0 is} sparse. Is that possible to recover both components? In this work, we prove that under certain suitable assumptions, we can recover both the low-rank and the sparse components exactly by simply solving a convex program whose objective is a weighted combination of the tensor nuclear norm and the l1-norm, i.e., min L, E ||L||* + λ||ε||1, s.t. X = L + ε, where λ = 1/√max(n₁, n₂)n₃. Interestingly, TRPCA involves RPCA as a special case when n₃ = 1 and thus it is a simple and elegant tensor extension of RPCA. Also numerical experiments verify our theory and the application for the image denoising demonstrates the effectiveness of our method.

Subfields: Computational Mathematics, Computational Mechanics, Signal Processing

View Paper: https://www.semanticscholar.org/paper/fcdf38dc8bd2d5ba23bd7d6ef2d00ea478d7cc4c

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mathematics

1. Yuan Xie
2. Shuicheng Yan
3. Dacheng Tao
4. Deyu Meng
5. Qi Xie
6. Pan Zhou
7. Chao Zhang
8. Zhenming Peng

In Computational Mechanics

1. Zhenming Peng
2. Landan Zhang
3. Dacheng Tao
4. Zhouchen Lin
5. Lei Zhang
6. Deyu Meng
7. Canyi Lu
8. Jiashi Feng

In Signal Processing

1. Zhenming Peng
2. Landan Zhang
3. Dacheng Tao
4. Zhouchen Lin
5. Lei Zhang
6. Deyu Meng
7. Qi Xie
8. Zongben Xu

Citations in the corpus

• Shared Predictive Cross-Modal Deep Quantization (2018, unknown): “eir private components. However, private components are coupled with shared components in the original representation. According to the idea of subspace decomposition [65] [66] and component analysis [67], explicitly modeling private components can enhance the ability to capture shared components ac”
• Infrared Small Target Detection Based on Partial Sum of the Tensor Nuclear Norm (2019, methodology): “…PSTNN [46] Partial Sum of Tensor Nuclear Norm t-SVD [47] Tensor Singular Value Decomposition RPCA [36] Robust Principle Component Analysis TRPCA [48] Tensor Robust Principle Component Analysis ADMM [49] Alternating Direction Method of Multipliers SNN [44] Sum of Nuclear Norms PSSV [50] Partial…”
• On Unifying Multi-view Self-Representations for Clustering by Tensor Multi-rank Minimization (2016, methodology): “2013; Kilmer andMartin 2011), which is referred to as t-SVD and has been applied to various tasks, such as image reconstruction and tensor completion (Semerci et al. 2014; Zhang et al. 2014; Lu et al. 2016).”
• Kronecker-Basis-Representation Based Tensor Sparsity and Its Applications to Tensor Recovery (2018, methodology): “When N ¼ 3 , it’s also comparable to the t-SVD-based TC/TRPCA methods [42], [74], which need to compute I 3 (assuming I 3 (cid:10) N ¼ 3 ) SVDs for I 1 (cid:2) I 2 matrices each iteration.”
• Tensor Robust Principal Component Analysis with a New Tensor Nuclear Norm (2018, methodology): “The above TNN definition is also different from existing works [19], [34], [27].”
• Tensor Factorization for Low-Rank Tensor Completion (2018, background): “Recent research [2], [3], [6], [7], [10]–[13] shows that high dimensional tensor data of interest, such as videos and image collections, are usually intrinsically low-rank or approximately so.”