[Paper Review] Dualization of Signal Recovery Problems

Patrick L. Combettes, Ðinh Dũng+1 more|arXiv (Cornell University)|Jul 2, 2009

Sparse and Compressive Sensing Techniques50 references4 citations

TL;DR

This paper introduces a novel duality framework for solving composite signal recovery problems via forward-backward splitting in the dual space, enabling strong convergence to the primal solution and weak convergence to the dual solution. The method generalizes existing algorithms, such as total variation denoising, and provides a unified approach applicable to a wide range of inverse problems in imaging and signal processing.

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.

Motivation & Objective

• To develop a unified duality framework for solving composite variational signal recovery problems that are difficult to solve via direct primal analysis.
• To leverage Fenchel-Moreau-Rockafellar duality to transform complex primal problems into more tractable dual problems amenable to forward-backward splitting.
• To ensure strong convergence of the primal iterates and weak convergence of the dual iterates, enhancing numerical reliability and solution accuracy.
• To extend existing duality-based signal recovery methods, such as those in total variation denoising and image restoration, to broader classes of problems.
• To provide a flexible algorithmic structure applicable to various inverse problems, including denoising, image reconstruction, and dictionary-based signal recovery.

Proposed method

• Formulate the primal signal recovery problem as a composite minimization involving a convex function and a linear operator, with constraints or regularization.
• Derive the Fenchel-Moreau-Rockafellar dual problem by conjugating the objective functions and transposing the linear operator.
• Apply forward-backward splitting to the dual problem, using proximity operators and iterative updates involving dual variables and step-sizes.
• Introduce a primal-dual iterative scheme where the dual variable is updated via projection onto a dual feasible set (e.g., ℓp*-ball), and the primal variable is reconstructed via proximity operator of the original function.
• Incorporate relaxation parameters (λn) and step-sizes (τn) to ensure convergence, with specific bounds on ε and μ to guarantee stability.
• Handle specific cases such as p=1, p=2, and p=∞ by using explicit projection formulas for the dual space, enabling efficient computation.

Experimental results

Research questions

• RQ1 Can duality theory be systematically applied to a broad class of composite signal recovery problems to simplify their solution?
• RQ2 Under what conditions does the dual problem admit a solution via forward-backward splitting, and how does this lead to convergence of the primal iterates?
• RQ3 How can the primal solution be recovered from the dual iterates, and what convergence guarantees (strong/weak) can be established?
• RQ4 To what extent can this framework generalize existing algorithms like total variation denoising and dictionary-based recovery methods?
• RQ5 What are the computational implications of using explicit projections in the dual space for different ℓp*-norms (e.g., p=1,2,∞)?

Key findings

• The proposed algorithm generates a sequence (vn) that converges weakly to a dual solution, and a sequence (xn) that converges strongly to the primal solution of the signal recovery problem.
• For the total variation denoising problem, the method reduces to known schemes when f=0, λn≡1, and τn≡τ∈]0,μ−1/4[, recovering the algorithm in [24].
• When p=1, the dual projection step becomes a soft-thresholding operation, allowing exact computation without error.
• For p=2, the projection onto the ℓ2-ball is computed via normalization of the dual variable, enabling efficient implementation.
• For p=∞, the method leverages an efficient algorithm for projecting onto the ℓ1-ball, as described in [10], ensuring computational efficiency.
• The convergence is guaranteed under the conditions that ∑n‖bn‖<∞ and ε∈]0,min{1,μ−1/8}], ensuring stability and convergence of the iterative scheme.