The number of singular vector tuples and uniqueness of best rank one approximation of tensors
This work provides foundational theoretical results for tensor decompositions, which are important for signal processing and data analysis, but the results are incremental as they extend known concepts from matrices to tensors.
The paper proves that a generic complex tensor has a finite number of singular vector tuples and provides a formula for this number. It also establishes the uniqueness of best rank-one approximations for almost all real tensors.
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.