[Paper Review] All Real Eigenvalues of Symmetric Tensors

Chunfeng Cui, Yu‐Hong Dai+1 more|arXiv (Cornell University)|Mar 14, 2014

Tensor decomposition and applications22 references4 citations

TL;DR

This paper proposes a novel sequential algorithm using Jacobian semidefinite relaxations in polynomial optimization to compute all real eigenvalues of symmetric tensors, from largest to smallest. It proves that each real eigenvalue can be computed via a finite hierarchy of semidefinite relaxations, and numerical experiments confirm its effectiveness, especially for cases where traditional methods like NSolve fail due to memory or time constraints.

ABSTRACT

This paper studies how to compute all real eigenvalues of a symmetric tensor. As is well known, the largest or smallest eigenvalue can be found by solving a polynomial optimization problem, while the other middle eigenvalues can not. We propose a new approach for computing all real eigenvalues sequentially, from the largest to the smallest. It uses Jacobian SDP relaxations in polynomial optimization. We show that each eigenvalue can be computed by solving a finite hierarchy of semidefinite relaxations. Numerical experiments are presented to show how to do this.

Motivation & Objective

• To develop a reliable method for computing all real eigenvalues of symmetric tensors, not just the extreme ones.
• To address the limitation of existing methods that cannot compute middle eigenvalues efficiently.
• To provide a finite, convergent hierarchy of semidefinite relaxations for computing all real eigenvalues in sequence.
• To demonstrate the method's scalability and robustness on various tensor sizes and orders.

Proposed method

• The method uses Lasserre’s hierarchy of semidefinite relaxations combined with Jacobian-based lifting to handle the eigenvalue problem as a polynomial optimization task.
• It formulates the computation of each real eigenvalue as a constrained polynomial optimization problem with equality and inequality constraints.
• A sequential approach is employed, starting from the largest eigenvalue and progressively computing smaller ones using deflation via a small positive parameter δ.
• The algorithm uses a finite hierarchy of semidefinite programs (SDPs) to approximate the real eigenvalues with increasing accuracy.
• The method incorporates symmetry and ordering constraints (e.g., $x_1 \geq x_2 \geq \cdots \geq x_n$) to improve numerical stability and convergence.
• It leverages the structure of symmetric tensors and the homogeneity of the form $\mathcal{A}x^m$ to reduce computational complexity.

Experimental results

Research questions

• RQ1 Can all real eigenvalues of a symmetric tensor be computed systematically, not just the largest or smallest?
• RQ2 How can a finite hierarchy of semidefinite relaxations be constructed to compute all real eigenvalues in sequence?
• RQ3 What is the computational scalability of the proposed method for higher-order and larger-dimensional tensors?
• RQ4 How does the method compare to direct solvers like NSolve in terms of time and memory efficiency?
• RQ5 What role does the parameter δ play in ensuring separation of closely spaced real eigenvalues?

Key findings

• For $n=4$, Algorithm 3.6 computes all four real Z-eigenvalues in 3.6 seconds, outperforming NSolve, which takes 19.3 seconds.
• For $n=5$, Algorithm 3.6 computes all five real Z-eigenvalues in 274.5 seconds, while NSolve fails to return results within 5 hours.
• For $n=8,9,10$, the algorithm computes the first three largest Z-eigenvalues but cannot compute smaller ones due to memory limits at relaxation order $N=4$.
• The algorithm successfully computes all real Z-eigenvalues for symmetric tensors with $n \leq 7$ and $m \leq 6$, as shown in Table 4.14.
• In a challenging case with $n=2$, $m=3$, and eigenvalues differing by $10^{-6}$, the algorithm correctly computes $\lambda_1 = 1.000001$, $\lambda_2 = 1.000000$, and $\lambda_3 = 0.707107$ in about 2 seconds.
• The method scales well for $n \leq 10$ and $m \leq 6$, but fails for larger $n$ or $m$ due to memory constraints at higher relaxation orders.