[Paper Review] Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs

Haibin Chen, Yannan Chen+2 more|arXiv (Cornell University)|Nov 7, 2015

Tensor decomposition and applications3 citations

TL;DR

This paper proposes a semidefinite programming algorithm to compute the maximum H-eigenvalue of even-order symmetric W-tensors—generalizing nonnegative and essentially nonnegative tensors—by leveraging a structured sum-of-squares (SOS) decomposition. The method enables polynomial-time computation and is applied to efficiently test copositivity of symmetric extended Z-tensors and compute Laplacian tensor eigenvalues in hypergraphs, with successful numerical results up to dimension 10,000.

ABSTRACT

Finding the maximum eigenvalue of a symmetric tensor is an important topic in tensor computation and numerical multilinear algebra. This paper is devoted to a semi-definite program algorithm for computing the maximum $H$-eigenvalue of a class of tensors with sign structure called $W$-tensors. The class of $W$-tensors extends the well-studied nonnegative tensors and essentially nonnegative tensors, and covers some important tensors arising naturally from spectral hypergraph theory. Our algorithm is based on a new structured sums-of-squares (SOS) decomposition result for a nonnegative homogeneous polynomial induced by a $W$-tensor. This SOS decomposition enables us to show that computing the maximum $H$-eigenvalue of an even order symmetric $W$-tensor is equivalent to solving a semi-definite program, and hence can be accomplished in polynomial time. Numerical examples are given to illustrate that the proposed algorithm can be used to find maximum $H$-eigenvalue of an even order symmetric $W$-tensor with dimension up to $10,000$. We present two applications for our proposed algorithm: we first provide a polynomial time algorithm for computing the maximum $H$-eigenvalues of large size Laplacian tensors of hyper-stars and hyper-trees; second, we show that the proposed SOS algorithm can be used to test the copositivity of a multivariate form associated with symmetric extended $Z$-tensors, whose order may be even or odd. Numerical experiments illustrate that our structured semi-definite program algorithm is effective and promising.

Motivation & Objective

• To develop a polynomial-time algorithm for computing the maximum H-eigenvalue of even-order symmetric W-tensors, a class extending nonnegative and essentially nonnegative tensors.
• To establish a structured sum-of-squares (SOS) decomposition for homogeneous polynomials induced by W-tensors, enabling semidefinite programming reformulation.
• To apply the proposed algorithm to test copositivity of symmetric extended Z-tensors of arbitrary even or odd order.
• To compute the maximum H-eigenvalues of Laplacian tensors for large hyper-stars and hyper-trees, which are important in spectral hypergraph theory.
• To demonstrate the scalability and effectiveness of the algorithm on high-dimensional tensors (up to dimension 10,000).

Proposed method

• Utilizes a novel structured sum-of-squares (SOS) decomposition for the homogeneous polynomial associated with a W-tensor, ensuring nonnegativity and enabling semidefinite programming reformulation.
• Reformulates the problem of computing the maximum H-eigenvalue of an even-order symmetric W-tensor as a semidefinite program (SDP), solvable in polynomial time.
• Employs the SOS decomposition to show that the maximum H-eigenvalue corresponds to the optimal value of a convex optimization problem over positive semidefinite matrices.
• Applies the algorithm to test copositivity of symmetric extended Z-tensors by analyzing the sign of the minimum value of a parameterized SOS program.
• Uses the YALMIP and SeDuMi MATLAB solvers to implement and test the semidefinite program on randomly generated tensors of varying orders and dimensions.
• Constructs extended Z-tensors with controlled diagonal dominance (via parameter M) to generate test instances and assess copositivity rates across different tensor sizes and orders.

Experimental results

Research questions

• RQ1 Can the maximum H-eigenvalue of an even-order symmetric W-tensor be computed efficiently, and is this problem solvable in polynomial time?
• RQ2 Does a structured sum-of-squares decomposition exist for the homogeneous polynomial associated with a W-tensor, enabling semidefinite programming reformulation?
• RQ3 Can the proposed algorithm be used to test the copositivity of symmetric extended Z-tensors of even or odd order?
• RQ4 What is the scalability of the algorithm in computing maximum H-eigenvalues for large-scale Laplacian tensors of hyper-stars and hyper-trees?
• RQ5 How does the parameter M in the construction of extended Z-tensors affect the observed copositivity rate in numerical experiments?

Key findings

• The maximum H-eigenvalue of an even-order symmetric W-tensor can be computed in polynomial time via semidefinite programming, thanks to a structured SOS decomposition.
• The algorithm successfully computes the maximum H-eigenvalue for tensors of dimension up to 10,000, demonstrating high scalability.
• For symmetric extended Z-tensors, the copositivity test via the proposed SOS-based SDP achieves a 99% success rate when the diagonal dominance parameter M is sufficiently large.
• The percentage of copositive instances increases with M: for m=3, n=2500, copositivity rises from 12% (M=11) to 99% (M=16); similar trends are observed across all tested orders and dimensions.
• The method enables efficient computation of the maximum H-eigenvalues of Laplacian tensors for hyper-stars and hyper-trees, which are otherwise difficult to compute due to their combinatorial structure.
• Numerical experiments confirm the algorithm's effectiveness and promise, with consistent convergence and high accuracy on large-scale instances.