The Number of Singular Vector Tuples and Uniqueness of Best Rank-One Approximation of Tensors

• Published in: Foundations of computational mathematics Vol. 14; no. 6; pp. 1209 - 1242
• Main Authors: Friedland, Shmuel, Ottaviani, Giorgio
• Format: Journal Article
• Language: English
• Published: Boston Springer US 01.12.2014; Springer; Springer Nature B.V
• Subjects: Analysis; Applications of Mathematics; Approximation; Approximation theory; Computer Science; Economics; Eigenvalues; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Numerical Analysis; Pencils; Tensors; Tensors (Mathematics); Texts; Uniqueness; Vector space; Vector spaces; Vectors (mathematics); United States; Singular vector tuples; Best rank-one approximation; approximation; 65K05; 14D21; 15A69; 15A18; Vector bundles; Chern classes; Partially symmetric tensors; Best rank; 65H10; Homogeneous pencil eigenvalue problem for cubic tensors; 65D15; Singular value decomposition
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-014-9194-z

In this paper we discuss the notion of singular vector tuples of a complex-valued d -mode tensor of dimension m 1 × ⋯ × m d . We show that a generic tensor has a finite number of singular vector tuples, viewed as points in the corresponding Segre product. We give the formula for the number of singular vector tuples. We show similar results for tensors with partial symmetry. We give analogous results for the homogeneous pencil eigenvalue problem for cubic tensors, i.e., m 1 = ⋯ = m d . We show the uniqueness of best approximations for almost all real tensors in the following cases: rank-one approximation; rank-one approximation for partially symmetric tensors (this approximation is also partially symmetric); rank- ( r 1 , … , r d ) approximation for d -mode tensors.