The computation of eigenvalues and eigenvectors is a cornerstone of linear algebra, with profound applications across science and engineering. The power method and the inverse power method are classical iterative algorithms designed to find the largest and the smallest eigenvalue respectively. In recent years, many problems in fields such as machine learning, data science, and image analysis have led to non-linear analogues of eigenvalue problems.
The project generalises the notion of eigenvectors and eigenvalues to the non-linear case in a suitable way, i.e. in such a way that the key properties from the linear case are preserved. Building upon this theoretical foundation, we propose novel generalisations of the power and inverse power methods tailored for non-linear settings. The central theoretical challenge lies in proving the convergence of these iterative schemes to the stationary points of the generalised Rayleigh quotient, despite the lack of convexity in general. By leveraging the Kurdyka-Łojasiewicz (KL) property as a core analytical tool, we overcome this limitation and establish rigorous convergence results for the proposed algorithms. This is joint work with Carola-Bibiane Schönlieb, Matthias Ehrhardt and Martin Benning.