A unifying Perron-Frobenius theorem for nonnegative tensors via multihomogeneous maps

Abstract

We introduce the concept of shape partition of a tensor and formulate a general tensor eigenvalue problem that includes all previously studied eigenvalue problems as special cases. We formulate irreducibility and symmetry properties of a nonnegative tensor $T$ in terms of the associated shape partition. We recast the eigenvalue problem for $T$ as a fixed point problem on a suitable product of projective spaces. This allows us to use the theory of multihomogeneous order-preserving maps to derive a new and unifying Perron–Frobenius theorem for nonnegative tensors which either implies earlier results of this kind or improves them, as weaker assumptions are required. We introduce a general power method for the computation of the dominant tensor eigenpair and provide a detailed convergence analysis.

Please cite this paper as:

@article{gautier2019unifying,
  title={A unifying {P}erron-{F}robenius theorem for nonnegative tensors via multihomogeneous maps},
  author={Gautier, Antoine and Tudisco, Francesco and Hein, Matthias},
  journal={SIAM Journal on Matrix Analysis and Applications},
  volume={40},
  number={3},
  pages={1206--1231},
  year={2019},
  publisher={SIAM}
}

Keywords: Perron-Frobenius theory nonlinear eigenvalues nonnegative tensors Hilbert metric Thompson metric power method multi-homogeneous maps