Talk by Francesco Tudisco

The Perron-Frobenius theory (1907-1912) provides fundamental properties about the eigensystems of nonnegative matrices. This theory has impacted many areas of mathematics, including graph theory, Markov chains, and matrix computation, and it forms a fundamental component in the analysis of a range of models in areas such as demography, economics, wireless networking, and search engine optimization. The nonlinear Perron-Frobenius theory allows us to transfer most of the theoretical and computational niceties of nonnegative matrices to the much broader class of nonlinear multihomogeneous operators. These types of operators include for example commonly used maps associated with tensors and are tightly connected to the formulation of nonlinear eigenvalue problems with eigenvector nonlinearities. In this talk, we will introduce the concept of multihomogeneous operators and we will present the state-of-the-art version of the nonlinear Perron-Frobenius theorem for nonnegative nonlinear mappings. We will compare this result with the classical theory for homogeneous operators and we will discuss several numerical optimization implications connected to nonlinear and higher-order versions of the Power and the Sinkhorn methods, including example problems in data science and machine learning which can be cast in terms of nonlinear eigenproblems. The talk is based on the recent SIREV SIGEST paper: Gautier, Tudisco, Hein, Nonlinear Perron-Frobenius theorems for nonnegative tensors, SIAM Review, Vol 65, pp 495–536, 2023.