Iterative Reweighted $\ell _{2}/\ell _{1}$ Recovery Algorithms for Compressed Sensing of Block Sparse Signals

In many applications of compressed sensing the signal is block sparse, i.e., the non-zero elements of the sparse signal are clustered in blocks. Here, we propose a family of iterative algorithms for the recovery of block sparse signals. These algorithms, referred to as iterative reweighted ℓ₂/ℓ₁ minimization algorithms (IR-ℓ₂/ℓ₁), solve a weighted ℓ₂/ℓ₁ minimization in each iteration. Our simulation and analytical results on the recovery of both ideally and approximately block sparse signals show that the proposed iterative algorithms have significant advantages in terms of accuracy and the number of required measurements over non-iterative approaches as well as existing iterative methods. In particular, we demonstrate that, by increasing the block length, the performance of the proposed algorithms approaches the Wu-Verdú theoretical limit. The improvement in performance comes at a rather small cost in complexity increase. Further improvement in performance is achieved by using a priori information about the location of non-zero blocks, even if such a priori information is not perfectly reliable.