Proximal Activation of Smooth Functions in Splitting Algorithms for Convex Image Recovery

TL;DR

This paper explores the efficiency of activating all functions proximally in splitting algorithms for convex image recovery, showing potential advantages over traditional gradient activation and providing new proximity operator examples.

Contribution

It introduces the concept of fully proximal activation in splitting algorithms and derives new proximity operators for smooth functions, enhancing computational efficiency in convex image recovery.

Findings

Fully proximal activation can outperform mixed approaches in certain scenarios.

New proximity operators for smooth convex functions are derived.

Numerical experiments demonstrate improved image recovery results.

Abstract

Structured convex optimization problems typically involve a mix of smooth and nonsmooth functions. The common practice is to activate the smooth functions via their gradient and the nonsmooth ones via their proximity operator. We show that, although intuitively natural, this approach is not necessarily the most efficient numerically and that, in particular, activating all the functions proximally may be advantageous. To make this viewpoint viable computationally, we derive a number of new examples of proximity operators of smooth convex functions arising in applications. A novel variational model to relax inconsistent convex feasibility problems is also investigated within the proposed framework. Several numerical applications to image recovery are presented to compare the behavior of fully proximal versus mixed proximal/gradient implementations of several splitting algorithms.