Finding the maximum eigenvalue of a class of tensors with applications in copositivity test and hypergraphs

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.