Proportionate affine projection algorithms from a basis pursuit perspective

It can be shown that the update of the affine projection algorithm (APA) can be decomposed as the sum of two orthogonal vectors. One of these vectors is derived from an ℓ₂-norm optimization problem while the other one is simply a good initialization vector. By replacing the ℓ₂-norm optimization with the ℓ_k-norm optimization (with 0 <; k ≤ 1), we can obtain a class of proportionate APAs (PAPAs). In this context, we evaluate the impact of the different ℓ_k-norm optimizations over the performance of the PAPAs.