Iterative Methods for Computing Eigenvectors of Nonlinear Operators

In this chapter we are examining several iterative methods for solving nonlinear eigenvalue problems. These arise in variational image processing, graph partition and classification, nonlinear physics, and more. The canonical eigenproblem we solve is T(u) = λu, where T : ℝ n → ℝ n $$T:\mathbb {R}^n\to \mathbb {R}^n$$ is some bounded nonlinear operator. Other variations of eigenvalue problems are also discussed. We present a progression of five algorithms, coauthored in recent years by the author and colleagues. Each algorithm attempts to solve a unique problem or to improve the theoretical foundations. The algorithms can be understood as nonlinear PDEs which converge to an eigenfunction in the continuous time domain. This allows a unique view and understanding of the discrete iterative process. Finally, it is shown how to evaluate numerically the results, along with some examples and insights related to priors of nonlinear denoisers, both classical algorithms and ones based on deep networks.