The Primal-Dual Hybrid Gradient Method for Semiconvex Splittings

TL;DR

This paper analyzes a primal-dual hybrid gradient method adapted for nonconvex regularizers, proving convergence under certain conditions and demonstrating its effectiveness through numerical experiments.

Contribution

It extends the primal-dual hybrid gradient method to semiconvex splittings, providing convergence analysis and practical validation for nonconvex variational problems.

Findings

Convergence is proven when nonconvexity is balanced by strong convexity.

Necessary and sufficient conditions for algorithm parameters are identified.

Numerical experiments confirm the method's effectiveness beyond theoretical guarantees.

Abstract

This paper deals with the analysis of a recent reformulation of the primal-dual hybrid gradient method [Zhu and Chan 2008, Pock, Cremers, Bischof and Chambolle 2009, Esser, Zhang and Chan 2010, Chambolle and Pock 2011], which allows to apply it to nonconvex regularizers as first proposed for truncated quadratic penalization in [Strekalovskiy and Cremers 2014]. Particularly, it investigates variational problems for which the energy to be minimized can be written as $G(u) + F(Ku)$, where $G$ is convex, $F$ semiconvex, and $K$ is a linear operator. We study the method and prove convergence in the case where the nonconvexity of $F$ is compensated by the strong convexity of the $G$. The convergence proof yields an interesting requirement for the choice of algorithm parameters, which we show to not only be sufficient, but necessary. Additionally, we show boundedness of the iterates under much weaker conditions. Finally, we demonstrate effectiveness and convergence of the algorithm beyond the theoretical guarantees in several numerical experiments.