A modified Riemannian Levenberg-Marquardt Algorithm for robust or constraint optimization on manifolds

We extend the Levenberg-Marquardt method on Riemannian manifolds to a robust variant that allows to tackle problems from applications where outliers are to be expected. We formally state the framework for a block-wise variant of the Robust Riemannian Levenberg-Marquardt algorithm and discuss how known convergence results can be applied here as well. We further discuss several alternatives for phrasing the sub problem that has to be solved. Finally we illustrate how the accompanying open source implementation in Manopt$.$jl can be used to efficiently solve problems such as geodesic regression, Procrustes analysis, subspace Procrustes analysis and bundle adjustment robustly and compare the Levenberg-Marquardt solver to other solvers for nonsmooth Riemannian optimization.