Tensor Golub–Kahan method based on Einstein product

• Published in: Journal of computational and applied mathematics Vol. 451; p. 116048
• Main Authors: El Hachimi, A., Jbilou, K., Hached, M., Ratnani, A.
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.12.2024
• Subjects: Einstein product; Tensor; Tensor Lanczos bidiagonalization; Tensor Ritz augmentation; Tensor singular triplets; Tensor SVD; Tensor singular triplets; Tensor; Einstein product; Tensor SVD; Tensor Ritz augmentation; Tensor Lanczos bidiagonalization
• ISSN: 0377-0427
• DOI: 10.1016/j.cam.2024.116048

The Singular Value Decomposition (SVD) of matrices is a widely used tool in scientific computing. In many applications of machine learning, data analysis, signal and image processing, the large datasets are structured into tensors, for which generalizations of SVD have already been introduced, for various types of tensor–tensor products. In this paper, we present innovative methods for approximating this generalization of SVD to tensors in the framework of the Einstein tensor product. These singular elements are called singular values and singular tensors respectively. The proposed method uses the tensor Lanczos bidiagonalization applied to the Einstein product. In most applications, as in the matrix case, the extremal singular values are of special interest. To enhance the approximation of the largest or the smallest singular triplets (singular values and left and right singular tensors), a restarted method based on Ritz augmentation is proposed. Numerical results are proposed to illustrate the effectiveness of the presented method. In addition, applications to video compression and facial recognition are presented.