Lower bounds on the rank and symmetric rank of real tensors. (September 2023)

• Record Type: Journal Article
• Authors: Wang, Kexin; Seigal, Anna
• Is Part Of: Journal of symbolic computation. Volume 118(2023)
• Page Start: 69
• Page End: 92
• Publication Date: 2023-09
• Subjects: 15A69 -- 14N07 -- 15A45 -- 15A30; Tensor decomposition -- Tensor rank -- Comon's conjecture; Mathematics -- Data processing -- Periodicals; Numerical analysis -- Data processing -- Periodicals; Automatic programming (Computer science) -- Periodicals; Mathématiques -- Informatique -- Périodiques; Analyse numérique -- Informatique -- Périodiques; Programmation automatique -- Périodiques; Automatic programming (Computer science); Mathematics -- Data processing; Numerical analysis -- Data processing; Periodicals; Electronic journals; 510.285
• DOI: 10.1016/j.jsc.2023.01.004
• Languages: English
• ISSNs: 0747-7171

Abstract

Abstract: We lower bound the rank of a tensor by a linear combination of the ranks of three of its unfoldings, using Sylvester's rank inequality. In a similar way, we lower bound the symmetric rank by a linear combination of the symmetric ranks of three unfoldings. Lower bounds on the rank and symmetric rank of tensors are important for finding counterexamples to Comon's conjecture. A real counterexample to Comon's conjecture is a tensor whose real rank and real symmetric rank differ. Previously, only one real counterexample was known, constructed in a paper of Shitov. We divide the construction into three steps. The first step involves linear spaces of binary tensors. The second step considers a linear space of larger decomposable tensors. The third step is to verify a conjecture that lower bounds the symmetric rank, on a tensor of interest. We use the construction to build an order six real tensor whose real rank and real symmetric rank differ.