Dualization of Signal Recovery Problems

TL;DR

This paper leverages duality in convex optimization to develop a forward-backward splitting algorithm for solving complex signal recovery problems, providing simultaneous convergence to primal and dual solutions and extending existing methods.

Contribution

It introduces a novel duality-based framework for composite variational problems in signal recovery, enabling efficient solution via forward-backward splitting and broadening applicability beyond current methods.

Findings

Algorithm converges weakly to dual solutions

Strong convergence to primal solutions

Framework extends existing duality-based methods

Abstract

In convex optimization, duality theory can sometimes lead to simpler solution methods than those resulting from direct primal analysis. In this paper, this principle is applied to a class of composite variational problems arising in particular in signal recovery. These problems are not easily amenable to solution by current methods but they feature Fenchel-Moreau-Rockafellar dual problems that can be solved by forward-backward splitting. The proposed algorithm produces simultaneously a sequence converging weakly to a dual solution, and a sequence converging strongly to the primal solution. Our framework is shown to capture and extend several existing duality-based signal recovery methods and to be applicable to a variety of new problems beyond their scope.