A randomized tensor singular value decomposition based on the t‐product

• Published in: Numerical linear algebra with applications Vol. 25; no. 5
• Main Authors: Zhang, Jiani, Saibaba, Arvind K., Kilmer, Misha E., Aeron, Shuchin
• Format: Journal Article
• Language: English
• Published: Oxford Wiley Subscription Services, Inc 01.10.2018
• Subjects: Computed tomography; Data compression; Datasets; Economic models; Face recognition; Image compression; Iterative methods; Randomization; randomized SVD; Singular value decomposition; tensor; Tensors; truncated SVD; t‐product
• ISSN: 1070-5325; 1099-1506
• DOI: 10.1002/nla.2179

Summary The tensor SVD (t‐SVD) for third‐order tensors, previously proposed in the literature, has been applied successfully in many fields, such as computed tomography, facial recognition, and video completion. In this paper, we propose a method that extends a well‐known randomized matrix method to the t‐SVD. This method can produce a factorization with similar properties to the t‐SVD, but it is more computationally efficient on very large data sets. We present details of the algorithms and theoretical results and provide numerical results that show the promise of our approach for compressing and analyzing image‐based data sets. We also present an improved analysis of the randomized and simultaneous iteration for matrices, which may be of independent interest to the scientific community. We also use these new results to address the convergence properties of the new and randomized tensor method as well.