Perturbation of higher-order singular values

Wolfgang Hackbusch, Daniel Kressner, and André Uschmajew

Contact the author: Please use for correspondence this email.
Submission date: 28. Jul. 2016
Pages: 16
published in: SIAM journal on applied algebra and geometry, 1 (2017) 1, p. 374-387 
DOI number (of the published article): 10.1137/16M1089873
Bibtex
MSC-Numbers: 15A18, 15A21, 15A69
Keywords and phrases: tensors, higher-order singular value decomposition, Newton method
Download full preprint: PDF (238 kB)

Abstract:
The higher-order singular values for a tensor of order d are defined as the singular values of the d different matricizations associated with the multilinear rank. When d ≥ 3, the singular values are generally different for different matricizations but not completely independent. Characterizing the set of feasible singular values turns out to be difficult. In this work, we contribute to this question by investigating which first-order perturbations of the singular values for a given tensor are possible. We prove that, except for trivial restrictions, any perturbation of the singular values can be achieved for almost every tensor with identical mode sizes. This settles a conjecture from [Hackbusch and Uschmajew, 2016] for the case of identical mode sizes. Our theoretical results are used to develop and analyze a variant of the Newton method for constructing a tensor with specified higher-order singular values or, more generally, with specified Gramians for the matricizations. We establish local quadratic convergence and demonstrate the robust convergence behavior with numerical experiments.