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

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 singul...