A fast and accurate algorithm for ℓ ₁ minimization problems in compressive sampling

An accurate and efficient algorithm for solving the constrained ℓ₁-norm minimization problem is highly needed and is crucial for the success of sparse signal recovery in compressive sampling. We tackle the constrained ℓ₁-norm minimization problem by reformulating it via an indicator function which describes the constraints. The resulting model is solved efficiently and accurately by using an elegant proximity operator-based algorithm. Numerical experiments show that the proposed algorithm performs well for sparse signals with magnitudes over a high dynamic range. Furthermore, it performs significantly better than the well-known algorithm NESTA (a shorthand for Nesterov’s algorithm) and DADM (dual alternating direction method) in terms of the quality of restored signals and the computational complexity measured in the CPU-time consumed.