Rate of Convergence of the FOCUSS Algorithm

Focal underdetermined system solver (FOCUSS) is a powerful method for basis selection and sparse representation, where it employs the ℓ
_p
-norm with p ∈ (0, 2) to measure the sparsity of solutions. In this paper, we give a systematical analysis on the rate of convergence of the FOCUSS algorithm with respect to p ∈ (0, 2). We prove that the FOCUSS algorithm converges superlinearly for 0 <; p <; 1 and linearly for 1 ≤ p <; 2 usually, but may superlinearly in some very special scenarios. In addition, we verify its rates of convergence with respect to p by numerical experiments.