3 years ago

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

Antoine Gautier, Francesco Tudisco, Matthias Hein

Inspired by the definition of symmetric decomposition, we introduce the concept of shape partition of a tensor and formulate a general tensor spectral problem that includes all the relevant spectral 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 spectral 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 general and unifying Perron-Frobenius theorem for nonnegative tensors that either implies previous results of this kind or improves them by weakening the assumptions there considered. We introduce a general power method for the computation of the dominant tensor eigenpair, and provide a detailed convergence analysis.

Publisher URL: http://arxiv.org/abs/1801.04215

DOI: arXiv:1801.04215v1

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