Sparse Recovery by Semi-Iterative Hard Thresholding Algorithm

Algorithm 1

Semi-iterative hard thresholding algorithm.

Step 1. Initialize 𝐱^[−1]=𝐱^[0]=𝐱₀

Step 2. Compute for 𝑘≥1, for a given 𝛾.

   𝜇₁=0,   𝜔₁=(4𝛾+2)/(4𝛾+1)

   𝜔_𝑘=42𝑘+2𝛾−1𝑘+𝛾−1/𝑘+2𝛾−12𝑘+2𝛾−1, 𝑘≥2

   𝜇_𝑘=(𝑘−1)(2𝑘−3)2𝑘+2𝛾−1/𝑘+2𝛾−12𝑘+4𝛾−12𝑘+2𝛾−3, 𝑘≥2

𝐚^[𝑘]=𝐱^[𝑘−1]+𝜇_𝑘(𝐱^[𝑘−1]−𝐱^[𝑘−2])+𝜔_𝑘Φ^T(𝑦−Φ𝐱^[𝑘−1])

Step 3. Compute 𝐱^[𝑘]=𝑃_𝑠(𝐚^[𝑘])

Step 4. Repeat step2-step3, until stopping criterion

   𝐲−Φ𝐱^[𝑘]₂≤𝜀‖𝐲‖₂ is satisfied.