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

arXiv: https://arxiv.org/abs/1801.04215 · HTML: https://mlsanity.com/paper/1801.04215/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jan 2018.

## Built on by (most cited first, 2 of 2)

- [You are AllSet: A Multiset Function Framework for Hypergraph Neural Networks](https://mlsanity.com/paper/2106.13264/) (2021)
- [Hypergraph Dissimilarity Measures](https://mlsanity.com/paper/2106.08206/) (2021)

## Abstract (arXiv)

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.