Iterative Methods for Computing Eigenvectors of Nonlinear Operators

TL;DR

This paper reviews iterative methods for solving nonlinear eigenvalue problems, highlighting five algorithms that converge to eigenfunctions and discussing their applications in image processing, physics, and graph analysis.

Contribution

Introduces and analyzes five iterative algorithms for nonlinear eigenvalue problems, providing a unified PDE-based convergence framework and practical evaluation methods.

Findings

Algorithms converge to eigenfunctions in continuous time.

Numerical evaluation methods are proposed.

Applications include image denoising and graph classification.

Abstract

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 $T(u)=\lambda u$, where $T:\R^n\to \R^n$ is some bounded nonlinear operator. Other variations of eigenvalue problems are also discussed. We present a progression of 5 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 PDE's 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.