Sparse Recovery Algorithms: Sufficient Conditions in terms of Restricted Isometry Constants

We review three recovery algorithms used in Compressive Sensing for the reconstruction s-sparse vectors x ∈ CN from the mere knowledge of linear measurements y = Ax ∈ Cm, m < N. For each of the algorithms, we derive improved conditions on the restricted isometry constants of the measurement matrix A that guarantee the success of the reconstruction. These conditions are δ2s < 0.4652 for basis pursuit, δ3s < 0.5 and δ2s < 0.25 for iterative hard thresholding, and δ4s < 0.3843 for compressive sampling matching pursuit. The arguments also applies to almost sparse vectors and corrupted measurements. The analysis of iterative hard thresholding is surprisingly simple. The analysis of basis pursuit features a new inequality that encompasses several inequalities encountered in Compressive Sensing.