Homotopy Methods Based on $l_{0}$ -Norm for Compressed Sensing

Area of Science:

• Signal Processing
• Numerical Analysis
• Optimization

Background:

• Compressed Sensing (CS) enables signal recovery from fewer measurements than traditional methods.
• Iterative Hard Thresholding (IHT) is a common algorithm for CS, but sensitive to regularization parameter choice.
• Determining optimal regularization parameters is crucial for effective CS recovery.

Purpose of the Study:

• To develop novel homotopy methods for solving the compressed sensing problem.
• To address the challenge of selecting regularization parameter values in IHT.
• To enhance the accuracy and efficiency of sparse signal recovery in CS.

Main Methods:

• Proposed two homotopy methods integrating the homotopy technique with Iterative Hard Thresholding (IHT).
• Analyzed the theoretical properties of the proposed methods, including convergence and sparsity bounds.
• Modified the methods into heuristic algorithms to improve solution quality.

Main Results:

• Demonstrated that accumulation points of the generated sequences are feasible solutions.
• Established an upper bound on the sparsity level for solutions.
• Computational experiments confirmed the effectiveness of the heuristic algorithms.

Conclusions:

• The proposed homotopy methods effectively solve the compressed sensing problem.
• The heuristic algorithms provide accurate and efficient sparse solutions, robust to noisy observations.
• These methods offer a practical advancement for compressed sensing applications.