The number of singular vector tuples and uniqueness of best rank one approximation of tensors

TL;DR

This paper investigates the properties and counts of singular vector tuples of complex tensors, explores their uniqueness in best rank-one approximations, and extends results to symmetric and partially symmetric tensors.

Contribution

It provides a formula for the number of singular vector tuples and establishes the generic uniqueness of best rank-one and low-rank approximations for tensors.

Findings

Finite number of singular vector tuples for generic tensors

Explicit formula for counting singular vector tuples

Almost sure uniqueness of best low-rank tensor approximations

Abstract

In this paper we discuss the notion of singular vector tuples of a complex valued $d$-mode tensor of dimension m_1 x ... x 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.