In this talk I will present a Perron-Frobenius type result for nonlinear eigenvector problems which allows us to compute the global maximum of a class of constrained nonconvex optimization problems involving multihomogeneous functions.
I will structure the talk into three main parts:
First, I will motivate the optimization of homogeneous functions from a graph partitioning point of view, showing an intriguing generalization of the famous Cheeger inequality.
Second, I will define the concept of multihomogeneous function and I will state our main Perron-Frobenious theorem. This theorem exploits the connection between optimization of multihomogeneous functions and nonlinear eigenvectors to provide an optimization scheme that has global convergence guarantees.
Third, I will discuss a few example applications in network science and machine learning that require the optimization of multihomogeneous functions and that can be solved using nonlinear Perron eigenvectors.