Minimization of Fraction Function Penalty in Compressed Sensing.

Area of Science:

• Signal Processing
• Optimization Theory
• Applied Mathematics

Background:

• Compressed sensing enables signal recovery from fewer measurements.
• Non-convex optimization poses challenges in signal recovery.
• Sparsity-promoting penalties are crucial for compressed sensing.

Purpose of the Study:

• To investigate the minimization of a non-convex fraction function penalty in compressed sensing.
• To establish the equivalence between l0 minimization and fraction function minimization.
• To develop and evaluate an iterative algorithm for sparse signal recovery.

Main Methods:

• Equivalence analysis between l0 minimization and fraction function minimization.
• Study of optimality conditions for the regularization problem.
• Derivation of a closed-form solution and development of an iterative FP thresholding algorithm.

Main Results:

• The optimal solution of the fraction function minimization solves the l0 minimization problem.
• The study derives properties of the optimal solution, including bounds on nonzero entries.
• The proposed FP algorithm demonstrates effective performance in sparse signal recovery.

Conclusions:

• The fraction function is a viable alternative to l0 minimization for sparse recovery.
• The iterative FP thresholding algorithm provides an efficient method for solving the minimization problem.
• The FP algorithm shows robustness in sparse signal recovery, even in the presence of noise.