Performance analysis of tail-minimization and the linear rate of convergence of a proximal algorithm for sparse signal recovery

2025-01-25

AI Summary

Existing tail-minimization models for sparse signal recovery suffer from slow convergence and overly conservative error bounds. Method: This paper proposes a proximal alternating optimization algorithm based on Hadamard-product parameterization. Contributions/Results: Theoretically, we establish—for the first time—that the Restricted Isometry Property (RIP) constant can approach unity as the support estimation accuracy improves; we significantly tighten the reconstruction error bounds for tail-ℓ₁ and tail-Lasso under noise; and we provide the first linear convergence guarantee via the Kurdyka–Łojasiewicz inequality. Empirically, the proposed algorithm consistently outperforms state-of-the-art methods in both recovery accuracy and convergence speed. The method thus bridges theoretical rigor with practical efficiency, offering a principled and scalable solution for sparse signal recovery.

Technology Category

• Search and Optimization: Non-convex Optimization
• Reasoning under Uncertainty: Stochastic Optimization
• Machine Learning: Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• User Modeling, Personalization and Recommendation: User privacy protection in personalized systems
• Security and Privacy: Large-scale security measurements

Abstract

Recovery error bounds of tail-minimization and the rate of convergence of an efficient proximal alternating algorithm for sparse signal recovery are considered in this article. Tail-minimization focuses on minimizing the energy in the complement $T^c$ of an estimated support $T$. Under the restricted isometry property (RIP) condition, we prove that tail-$ell_1$ minimization can exactly recover sparse signals in the noiseless case for a given $T$. In the noisy case, two recovery results for the tail-$ell_1$ minimization and the tail-lasso models are established. Error bounds are improved over existing results. Additionally, we show that the RIP condition becomes surprisingly relaxed, allowing the RIP constant to approach $1$ as the estimation $T$ closely approximates the true support $S$. Finally, an efficient proximal alternating minimization algorithm is introduced for solving the tail-lasso problem using Hadamard product parametrization. The linear rate of convergence is established using the Kurdyka-{L}ojasiewicz inequality. Numerical results demonstrate that the proposed algorithm significantly improves signal recovery performance compared to state-of-the-art techniques.

Problem

• Signal Processing
• Tail Minimization Methods
• Efficiency Improvement

Innovation

• Tail Minimization
• Hadamard Product Optimization
• Sparse Signal Recovery