joint work with Roland Herzog, Daniel Tenbrinck, Jose Vidal-Nunez
Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.
joint work with Duy Hoang Thai, Stephan Huckemann
Removing texture from an image while preserving sharp edges, to obtain a so-called cartoon, is a challenging task in image processing, especially since the term texture is mathematically not well defined. We propose a generalized family of algorithms, guided by two optimization problems simultaneously, profiting from the advantages of both variational calculus and harmonic analysis. In particular, we aim at avoiding introduction of artefacts and staircasing, while keeping the contrast of the image.
joint work with Felix Krahmer
Matrix completion refers to the problem of reconstructing a low-rank matrix from only a random subset of its entries. It has been studied intensively in recent years due to various applications in learning and data science. A benchmark algorithm for this problem is nuclear norm minimization, a natural convex proxy for the rank. We will discuss the stability of this approach under the assumption that the observations are perturbed by adversarial noise.