[Paper Review] Circulant Tensors with Applications to Spectral Hypergraph Theory and Stochastic Process

Zhongming Chen, Liqun Qi|arXiv (Cornell University)|Dec 10, 2013

Tensor decomposition and applications14 references16 citations

TL;DR

This paper establishes spectral properties of circulant tensors, showing that the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified, and proves that even-order circulant B₀ tensors are positive semi-definite. These results confirm structural properties in circulant hypergraphs and directed circulant hypergraphs, and extend to mth-order stationary stochastic processes, where their mth-order moments (as circulant tensors) are positive semi-definite when m is even.

ABSTRACT

Circulant tensors naturally arise from stochastic process and spectral hypergraph theory. The joint moments of stochastic processes are symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of circulant hypergraphs are also symmetric circulant tensors. The adjacency, Laplacian and signless Laplacian tensors of directed circulant hypergraphs are circulant tensors, but they are not symmetric in general. In this paper, we study spectral properties of circulant tensors and their applications in spectral hypergraph theory and stochastic process. We show that in certain cases, the largest H-eigenvalue of a circulant tensor can be explicitly identified. In particular, the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified. This confirms the results in circulant hypergraphs and directed circulant hypergraphs. We prove that an even order circulant B$_0$ tensor is always positive semi-definite. This shows that the Laplacian tensor and the signless Laplacian tensor of a directed circulant even-uniform hypergraph are positive semi-definite. If a stochastic process is $m$th order stationary, where $m$ is even, then its $m$th order moment, which is a circulant tensor, must be positive semi-definite. In this paper, we give various conditions for a circulant tensor to be positive semi-definite.

Motivation & Objective

• To investigate spectral properties of circulant tensors arising in stochastic processes and spectral hypergraph theory.
• To establish conditions under which even-order circulant tensors are positive semi-definite.
• To confirm that the largest H-eigenvalue of a nonnegative circulant tensor can be explicitly identified.
• To prove that even-order circulant B₀ tensors are positive semi-definite, extending to Laplacian and signless Laplacian tensors of directed circulant hypergraphs.
• To show that mth-order stationary stochastic processes with even m have positive semi-definite mth-order moment tensors.

Proposed method

• Define circulant tensors via invariance under cyclic shifts of indices, generalizing circulant matrices to higher-order tensors.
• Introduce the root tensor and associated tensor to generate any circulant tensor from its first slice.
• Use the discrete Fourier transform (DFT) framework to derive explicit eigenvalues and eigenvectors for circulant tensors.
• Formulate the minimization of the tensor multilinear form under unit norm constraints as a constrained optimization problem.
• Apply the alternating direction method of multipliers (ADMM) to solve the tensor optimization problem with convergence guarantees under certain conditions.
• Implement ADMM with closed-form subproblem solutions for the unit ball constraint, enabling efficient numerical computation.

Experimental results

Research questions

• RQ1 Can the largest H-eigenvalue of a nonnegative circulant tensor be explicitly identified?
• RQ2 Under what conditions is an even-order circulant tensor positive semi-definite?
• RQ3 Are the Laplacian and signless Laplacian tensors of directed circulant hypergraphs positive semi-definite?
• RQ4 Is the mth-order moment tensor of an mth-order stationary stochastic process positive semi-definite when m is even?
• RQ5 Can the optimization of the tensor multilinear form over the unit sphere be efficiently solved using ADMM?

Key findings

• The largest H-eigenvalue of a nonnegative circulant tensor can be explicitly computed, confirming spectral properties in circulant hypergraphs.
• Every even-order circulant B₀ tensor is positive semi-definite, implying that Laplacian and signless Laplacian tensors of directed circulant even-uniform hypergraphs are positive semi-definite.
• For any mth-order stationary stochastic process with even m, the mth-order moment tensor—a circulant tensor—is positive semi-definite.
• The ADMM algorithm converges to the global minimum with 100% success rate in 100 trials for tested examples (n=3,4), achieving accuracy within 10⁻⁵ of the true solution.
• Numerical results show the ADMM method is efficient, with average convergence in 62.73 and 92.49 iterations for 4th-order tensors of size 3 and 4, respectively.
• The algorithm remains effective for small-scale problems, though scalability is limited by the inability of Nsolve to handle larger instances.