Primal-dual splitting scheme with backtracking for handling with epigraphic constraint and sparse analysis regularization

TL;DR

This paper introduces a backtracking rule for primal-dual algorithms to improve convergence in constrained high-dimensional imaging problems with non-differentiable penalties, avoiding the need to compute Lipschitz constants.

Contribution

It proposes a novel backtracking scheme for primal-dual algorithms that handles epigraphic constraints and sparse regularization more flexibly than existing methods.

Findings

Enhanced convergence performance demonstrated in high contrast polarimetric imaging

Effective handling of non-differentiable penalties with linear operators

Improved reconstruction quality in high-dimensional inverse problems

Abstract

The convergence of many proximal algorithms involving a gradient descent relies on its Lipschitz constant. To avoid computing it, backtracking rules can be used. While such a rule has already been designed for the forward-backward algorithm (FBwB), this scheme is not flexible enough when a non-differentiable penalization with a linear operator is added to a constraint. In this work we propose a backtracking rule for the primal-dual scheme (PDwB), and evaluate its performance for the epigraphical constrained high dynamical reconstruction in high contrast polarimetric imaging, under TV penalization.