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

Abstract

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 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.2010 Mathematics Subject Classification. 14D21, 15A18, 15A69, 65D15, 65H10, 65K05.

Tensors : A brief introduction

“…This problem may be seen to be related to tensor eigenvalues [17] [53] [59] [35] [88]. It has been proved recently that the best rank-1 approximation of a symmetric tensor is symmetric [34]; a shorter proof can be found in [35], as well as uniqueness issues. So a question deserves to be raised: can the exact or approximate CP decompositions be computed by successive rank-1 approximations?…”

Low-Rank Approximation of Tensors

“…Then the critical points of f T |S (n) are given by the Lagrange multipliers formulas [33]: 3) is bounded by the numbers given in [17].) Consider first the maximization problem of f T (x 1 , .…”

The euclidean distance degree

“…Work of Catanese and Trifogli [4] offers degree formulas for ED discriminants in various situations. Section 8 covers the approximation of tensors by rank one tensors, along the lines of [8,9].…”