[Paper Review] ADMM-MCP Framework for Sparse Recovery with Global Convergence

Hao Wang, Zhang-Lei Shi+2 more|arXiv (Cornell University)|May 2, 2018

Sparse and Compressive Sensing Techniques23 references3 citations

TL;DR

This paper proposes an ADMM-MCP framework for sparse signal recovery in compressed sensing that achieves global convergence by combining the minimax concave penalty (MCP) with the alternating direction method of multipliers (ADMM) and a modified iterative hard thresholding (IHT) scheme. The method ensures convergence to a globally optimal solution under mild assumptions and outperforms state-of-the-art algorithms in reconstruction accuracy with fewer measurements.

ABSTRACT

In compressed sensing, the l0-norm minimization of sparse signal reconstruction is NP-hard. Recent work shows that compared with the best convex relaxation (l1-norm), nonconvex penalties can better approximate the l0-norm and can reconstruct the signal based on fewer observations. In this paper, the original problem is relaxed by using minimax concave penalty (MCP). Then alternating direction method of multipliers (ADMM) and modified iterative hard thresholding method are used to solve the problem. Under certain reasonable assumptions, the global convergence of the proposed method is proved. The parameter setting is also discussed. Finally, through simulations and comparisons with several state-of-the-art algorithms, the effectiveness of proposed method is confirmed.

Motivation & Objective

• To address the NP-hard nature of $l_0$-norm minimization in sparse signal recovery by replacing it with a nonconvex penalty that better approximates sparsity.
• To overcome the limitations of $l_1$-norm relaxation, which requires more measurements and imposes strict conditions for exact recovery.
• To develop a globally convergent algorithm that avoids dependence on initialization, unlike existing iterative hard thresholding methods.
• To combine the strengths of ADMM (for decomposition and convergence) and MCP (for unbiased, nonconvex sparsity promotion) in a unified framework.
• To provide a theoretically grounded parameter selection strategy for the regularization parameter $\lambda$.

Proposed method

• The original $l_0$-norm minimization problem is relaxed using the minimax concave penalty (MCP), which provides a nonconvex, continuous approximation to sparsity.
• The ADMM framework is employed to decompose the problem into subproblems that are solved iteratively with dual ascent on the augmented Lagrangian.
• A modified iterative hard thresholding (IHT) step is integrated into the ADMM scheme to enforce sparsity at each iteration by retaining only the largest $\tau$ components.
• The algorithm alternates between updating the primal variables $\boldsymbol{x}$, dual variables $\boldsymbol{u}$, and the auxiliary variable $\boldsymbol{w}$, ensuring convergence under mild assumptions.
• The method incorporates a line search strategy for the penalty parameter $\lambda$ to balance data fidelity and sparsity, with two parameter selection rules analyzed.

Experimental results

Research questions

• RQ1 Can a nonconvex penalty like MCP be effectively combined with ADMM to achieve global convergence in sparse recovery?
• RQ2 Does the proposed ADMM-MCP framework outperform convex $l_1$-based methods and other nonconvex solvers in terms of reconstruction accuracy and measurement efficiency?
• RQ3 Can the modified IHT step within ADMM ensure global convergence despite the nonconvexity of the MCP penalty?
• RQ4 How sensitive is the performance to the choice of the regularization parameter $\lambda$, and can automated selection strategies be effective?
• RQ5 What is the convergence behavior of the algorithm under varying noise levels and sparsity levels?

Key findings

• The proposed ADMM-MCP method achieves global convergence under mild assumptions, eliminating the need for careful initialization.
• The method requires significantly fewer measurements than $l_1$-based methods to achieve the same probability of exact reconstruction, as shown in simulations with $N=512$ and $N=4096$.
• With the first parameter selection method, the proposed algorithm achieves near-optimal performance, outperforming SPGL1, FOCUSS, and other nonconvex solvers under high noise.
• The second parameter selection method, while less accurate than the first, still yields better performance than most competing algorithms and enables automatic tuning.
• The algorithm converges within 100–150 iterations even under low signal-to-noise ratios ($\sigma = 0.01$), demonstrating robustness and efficiency.
• The method is robust to initialization, as demonstrated by successful recovery when starting from $\boldsymbol{x}^0 = \mathbf{0}$, $\boldsymbol{u}^0 = \mathbf{0}$, and $\boldsymbol{w}^0 = \mathbf{0}$.