Forward-backward splitting in bilaterally bounded Alexandrov spaces

Abstract

With the goal of solving optimisation problems on non-Riemannian manifolds, such as geometrical surfaces with sharp edges, we develop and prove the convergence of a forward-backward method in Alexandrov spaces with curvature bounded both from above and from below. This bilateral boundedness is crucial for the availability of both the gradient and proximal steps, instead of just one or the other. We numerically demonstrate the behaviour of the proposed method on simple geometrical surfaces in $\mathbb{R}^3$.