Francesco Tudisco

Associate Professor (Reader) in Machine Learning

School of Mathematics, The University of Edinburgh
The Maxwell Institute for Mathematical Sciences
School of Mathematics, Gran Sasso Science Institute JCMB, King’s Buildings, Edinburgh EH93FD UK
email: f dot tudisco at

Nonlinear Perron-Frobenius theorems for nonnegative tensors

Antoine Gautier, Francesco Tudisco, Matthias Hein,
SIAM Review (SIGEST paper), 65 : 495--536 (2023)


We present a unifying Perron–Frobenius theory for nonlinear spectral problems defined in terms of nonnegative tensors. By using the concept of tensor shape partition, our results include, as a special case, a wide variety of particular tensor spectral problems considered in the literature and can be applied to a broad set of problems involving tensors (and matrices), including the computation of operator norms, graph and hypergraph matching in computer vision, hypergraph spectral theory, higher-order network analysis, and multimarginal optimal transport. The key to our approach is to recast the eigenvalue problem 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 new and unifying Perron–Frobenius theorems for nonnegative tensors, which either imply earlier results of this kind or improve 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. This paper is directly based on our previous work [A. Gautier, F. Tudisco, and M. Hein, SIAM J. Matrix Anal. Appl., 40 (2019), pp. 1206–1231] and complements it by providing an extended introduction and several new results.

Please cite this paper as:

	author = {Gautier, Antoine and Tudisco, Francesco and Hein, Matthias},
	doi = {10.1137/23M1557489},
	journal = {SIAM Review},
	number = {2},
	pages = {495-536},
	title = {Nonlinear Perron--Frobenius Theorems for Nonnegative Tensors},
	volume = {65},
	year = {2023}

Links: doi

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