[Paper Review] Exclusion sets in eigenvalue inclusion sets for tensors

Chaoqian Li, Suhua Li+2 more|arXiv (Cornell University)|Jun 3, 2017

Tensor decomposition and applications13 references3 citations

TL;DR

This paper proposes two new eigenvalue inclusion sets, Ω(𝒜) and Θ(𝒜), by excluding subsets that contain no eigenvalues from the Geršgorin set Γ(𝒜) and Brauer-type set 𝒦(𝒜), respectively. The key contribution is that Ω(𝒜) ⊆ Γ(𝒜) and Θ(𝒜) ⊆ 𝒦(𝒜), providing tighter bounds for tensor eigenvalues using eigenvector-based exclusion criteria.

ABSTRACT

By excluding some sets, which don't include any eigenvalue of a tensor, from some existing eigenvalue inclusion sets, two new sets are given to locate all eigenvalues of a tensor. And it is shown that these two sets are contained in the Geršgorin eigenvalue inclusion set of tensors provide by Qi (Journal of Symbolic Computation 2005; 40:1302-1324) and the Brauer-type eigenvalue inclusion set provide by Li et al. (Numer. Linear Algebra Appl. 2014; 21:39-50) respectively. Two sufficient conditions such that the determinant of a tensor is not zero are also provided.

Motivation & Objective

• To improve existing eigenvalue inclusion sets for tensors by identifying and excluding regions that contain no eigenvalues.
• To refine the Geršgorin set Γ(𝒜) and Brauer-type set 𝒦(𝒜) by removing subsets that do not include any eigenvalues of a tensor.
• To develop new, tighter eigenvalue inclusion sets Ω(𝒜) and Θ(𝒜) that are strictly contained within Γ(𝒜) and 𝒦(𝒜), respectively.
• To provide sufficient conditions for a tensor's determinant to be non-zero based on the exclusion of eigenvalue-free regions.
• To extend the applicability of eigenvalue localization techniques to higher-order tensors using eigenvector modulus and component analysis.

Proposed method

• Define exclusion sets Δᵢ(𝒜) and Λᵢ(𝒜) that remove eigenvalue-free regions from the Geršgorin and Brauer-type sets, respectively.
• Use the eigenvector's maximum modulus component and the corresponding tensor equation to derive bounds on eigenvalue locations.
• Construct Ωᵢ(𝒜) = Γᵢ(𝒜) \ Δᵢ(𝒜) by excluding Δᵢ(𝒜), a region defined via inequalities involving |xₚ| and |xₜ|, to refine the Geršgorin disk.
• Formulate Θᵢⱼ(𝒜) = 𝒦ᵢⱼ(𝒜) \ Λᵢ(𝒜) by removing Λᵢ(𝒜), a union of sets defined by inequalities involving |λ - aᵢᵢ|, |λ - aₚₚ|, and coefficients aᵢₚ…ₚ.
• Derive exclusion conditions using triangle inequality and component-wise bounds on eigenvector entries to eliminate regions where eigenvalues cannot lie.
• Establish inclusion relationships Ω(𝒜) ⊆ Γ(𝒜) and Θ(𝒜) ⊆ 𝒦(𝒜) through case analysis on eigenvector components and modulus comparisons.

Experimental results

Research questions

• RQ1 Which subsets of the Geršgorin set Γ(𝒜) do not contain any eigenvalues of a tensor, and how can they be systematically excluded?
• RQ2 Can the Brauer-type eigenvalue inclusion set 𝒦(𝒜) be tightened by removing eigenvalue-free regions based on eigenvector structure?
• RQ3 What conditions on tensor entries ensure that certain regions in Γ(𝒜) or 𝒦(𝒜) are guaranteed to exclude all eigenvalues?
• RQ4 How can the exclusion of such regions lead to improved determinant criteria for non-singularity of tensors?
• RQ5 To what extent do the new sets Ω(𝒜) and Θ(𝒜) improve upon existing inclusion sets in terms of eigenvalue localization accuracy?

Key findings

• The set Ω(𝒜) is constructed by excluding Δᵢ(𝒜) from each Geršgorin disk Γᵢ(𝒜), resulting in Ω(𝒜) ⊆ Γ(𝒜), and it strictly excludes regions that cannot contain eigenvalues.
• The set Θ(𝒜) is formed by removing Λᵢ(𝒜) from the Brauer-type set 𝒦(𝒜), yielding Θ(𝒜) ⊆ 𝒦(𝒜), and it provides a tighter inclusion region than 𝒦(𝒜).
• For any eigenvalue λ of a tensor, λ ∉ Λᵢ(𝒜) for all i, meaning the sets Λᵢ(𝒜) are valid exclusion sets for the Brauer-type inclusion set.
• The exclusion sets Δᵢ(𝒜) and Λᵢ(𝒜) are derived from eigenvector component analysis and inequalities involving |xₜ| and |xₚ|, ensuring they contain no eigenvalues.
• Corollary 3 provides two sufficient conditions for det(𝒜) ≠ 0 based on coefficient inequalities involving |aᵢᵢ|, rᵢ(𝒜), and off-diagonal entries.
• The new sets Ω(𝒜) and Θ(𝒜) are strictly contained within Γ(𝒜) and 𝒦(𝒜), respectively, and are not necessarily contained within each other, indicating complementary improvements.