Francesco Tudisco: Computing nonlinear Perron eigenvectors of positive matrices and tensors

Abstract:

Tensors with nonnegative entries arise very naturally in numerous applications and, as for the matrix case, one often is interested in their dominant positive eigen and singular vectors. The power method is the algorithm of reference for this problems. However, unlike the matrix case, most of the eigenvector problems for tensors are NP-hard and thus the convergence of this method cannot be guaranteed in general. In this talk I will discuss a new condition that ensures existence, uniqueness and computability of the nonlinear Perron eigenvector for a large class of positive matrices and tensors. This result is based on a multilinear version of the famous Birkhoff-Hopf theorem for matrices that we have recently proved. In particular, I will list a number of nonlinear spectral problems for positive matrices and tensors and, for all of them, I will detail the corresponding conditions that guarantee the global convergence of the associated nonlinear power method.