A unifying Perron-Frobenius theorem for nonnegative tensors via multi-homogeneous 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 multi-homogeneous 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.

arXiv commentauthor's accepted version

References in corpus (5)

• Perron-Frobenius theorem for nonnegative multilinear forms and extensions
• Shifted Power Method for Computing Tensor Eigenpairs
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• Gradient Flows and Nonlinear Power Methods for the Computation of Nonlinear Eigenfunctions
• On Limiting Probability Distributions of Higher Order Markov Chains