A faster tensor robust PCA via tensor factorization
- PDF / 13,247,757 Bytes
- 21 Pages / 595.276 x 790.866 pts Page_size
- 65 Downloads / 230 Views
ORIGINAL ARTICLE
A faster tensor robust PCA via tensor factorization An‑Dong Wang1,2 · Zhong Jin1,2 · Jing‑Yu Yang1,2 Received: 2 December 2019 / Accepted: 9 June 2020 © Springer-Verlag GmbH Germany, part of Springer Nature 2020
Abstract Many kinds of real-world multi-way signal, like color images, videos, etc., are represented in tensor form and may often be corrupted by outliers. To recover an unknown signal tensor corrupted by outliers, tensor robust principal component analysis (TRPCA) serves as a robust tensorial modification of the fundamental PCA. Recently, a successful TRPCA model based on the tubal nuclear norm (TNN) (Lu et al. in IEEE Trans Pattern Anal Mach Intell 42:925–938, 2019) has attracted much attention thanks to its superiority in many applications. However, TNN is computationally expensive due to the requirement of full singular value decompositions, seriously limiting its scalability to large tensors. To address this issue, we propose a new TRPCA model which adopts a factorization strategy. Algorithmically, an algorithm based on the non-convex augmented Lagrangian method is developed with convergence guarantee. Theoretically, we rigorously establish the sub-optimality of the proposed algorithm. We also extend the proposed model to the robust tensor completion problem. Both the effectiveness and efficiency of the proposed algorithm is demonstrated through extensive experiments on both synthetic and real data sets.
1 Introduction PCA is arguably the most broadly applied statistical approach for high-dimensional data analysis and dimension reduction. However, it regards each data instance as a vector, ignoring the rich intra-mode and inter-mode information in the emerging multi-way data (tensor data). One the other hand, it is sensitive to outliers which are ubiquitous in real applications. By manipulating the tensor instance in its original multi-way form and attempting to work well against outliers, tensor robust PCA [7, 23] is a powerful extension of PCA which can overcome the above issues. TRPCA finds many real life applications likes image/video restoration, video surveillance, face recognition, to name a few [7, 23].
Part of this work [32] has been presented in 5th IAPR Asian Conference on Pattern Recognition (ACPR), 26-29 November 2019, Auckland, New Zealand. * Zhong Jin [email protected] 1
School of Computer Science and Engineering, Nanjing University of Science and Technology, Nanjing 210094, China
Key Laboratory of Intelligent Perception and System for High‑Dimensional Information of Ministry of Education, Nanjing University of Science and Technology, Nanjing 210094, China
2
As shown in Fig. 1, an idealized version of TRPCA aims to recover an underlying tensor L∗ from measurements M corrupted by outliers represented by tensor S∗ , that is,
M = L∗ + S∗ .
(1)
Obviously, the decomposition in Model (1) is impossible without additional assumptions on the underlying tensor L∗ and the outlier tensor S∗ . Thus, TRPCA further assumes L∗ is “low-rank” and S∗ sparse. Mathemat
Data Loading...
