[Paper Review] Eigenvalues and structured properties of P-tensors

Yisheng Song, Liqun Qi|arXiv (Cornell University)|Aug 9, 2015

Tensor decomposition and applications27 references3 citations

TL;DR

This paper introduces two new constants, δ_H(𝒜) and δ_Z(𝒜), based on H- and Z-eigenvalues of P-tensors, and establishes tight upper bounds for the key quantities α(F_𝒜) and α(T_𝒜) that characterize P-tensor properties. For even-order P-tensors, it proves α(F_𝒜) ≤ (δ_H(𝒜))^{1/(m−1)} and α(T_𝒜) ≤ δ_Z(𝒜), both bounded above by diagonal entries, providing a quantitative link between eigenvalues and P-tensor structure.

ABSTRACT

We define two new constants associated with real eigenvalues of a P-tensor. With the help of these two constants, in the case of P-tensors, we establish upper bounds of two important quantities, whose positivity is a necessary and sufficient condition for a general tensor to be a P-tensor.

Motivation & Objective

• To define new constants δ_H(𝒜) and δ_Z(𝒜) based on H- and Z-eigenvalues of P-tensors to analyze their structural properties.
• To establish upper bounds for α(F_𝒜) and α(T_𝒜), which are necessary and sufficient conditions for a tensor to be a P-tensor.
• To provide eigenvalue-based quantitative estimates that link the diagonal entries of a P-tensor to its P-property, especially for even-order tensors.
• To address open questions on the lower bounds and tightness of these upper bounds for α(F_𝒜) and α(T_𝒜).

Proposed method

• Define δ_H(𝒜) as the minimum H-eigenvalue over all principal sub-tensors 𝒜_r^J of 𝒜.
• Define δ_Z(𝒜) as the minimum Z-eigenvalue over all principal sub-tensors 𝒜_r^J of 𝒜.
• Use variational characterization of H- and Z-eigenvalues to derive inequalities involving the real vector y with ∥y∥_∞ = 1.
• Apply the definition of α(F_𝒜) = max_i y_i (F_𝒜(y))_i and α(T_𝒜) = max_i y_i (T_𝒜(y))_i for unit infinity-norm vectors.
• Establish upper bounds via inequalities involving y_i(𝒜y^{m−1})_i ≤ δ ∥y∥_∞^{m−2} ∥y∥_2^2, leading to α(F_𝒜) ≤ δ_H(𝒜)^{1/(m−1)} and α(T_𝒜) ≤ δ_Z(𝒜).
• Leverage known results on P-tensors and the fact that 𝒜 is a P-tensor iff α(T_𝒜) > 0 (for even m) or α(F_𝒜) > 0 (for even m), to validate the bounds.

Experimental results

Research questions

• RQ1 Do the quantities α(F_𝒜) and α(T_𝒜) have a positive lower bound for P-tensors?
• RQ2 Are the derived upper bounds for α(F_𝒜) and α(T_𝒜) the tightest possible?
• RQ3 Can the constants δ_H(𝒜) and δ_Z(𝒜) be used to characterize the P-tensor property in terms of eigenvalues?
• RQ4 How do the diagonal entries of a P-tensor constrain the values of α(F_𝒜) and α(T_𝒜)?
• RQ5 What is the relationship between the H- and Z-eigenvalues of principal sub-tensors and the P-tensor property?

Key findings

• For an even-order P-tensor 𝒜, the quantity α(F_𝒜) is bounded above by (δ_H(𝒜))^{1/(m−1)}, where δ_H(𝒜) is the minimum H-eigenvalue over all principal sub-tensors.
• For an even-order P-tensor 𝒜, the quantity α(T_𝒜) is bounded above by δ_Z(𝒜), the minimum Z-eigenvalue over all principal sub-tensors.
• The upper bound δ_H(𝒜) ≤ min_i a_{i⋯i} holds, so α(F_𝒜) ≤ (min_i a_{i⋯i})^{1/(m−1)} for even m.
• The upper bound δ_Z(𝒜) ≤ min_i a_{i⋯i} holds, so α(T_𝒜) ≤ min_i a_{i⋯i} for even m.
• The bounds are tight in the sense that they depend only on the diagonal entries of the tensor and are derived from eigenvalue-based constants.
• The results confirm that α(F_𝒜) > 0 and α(T_𝒜) > 0 are necessary and sufficient for a tensor to be a P-tensor when m is even, and the bounds provide quantitative control over these values.