On recovery of block-sparse signals via mixed l ₂ /l _q (0 < q ≤ 1)norm minimization

Compressed sensing (CS) states that a sparse signal can exactly be recovered from very few linear measurements. While in many applications, real-world signals also exhibit additional structures aside from standard sparsity. The typical example is the so-called block-sparse signals whose non-zero coefficients occur in a few blocks. In this article, we investigate the mixed l ₂/l _q (0 < q ≤ 1) norm minimization method for the exact and robust recovery of such block-sparse signals. We mainly show that the non-convex l ₂/l _q (0 < q < 1) minimization method has stronger sparsity promoting ability than the commonly used l ₂/l ₁ minimization method both practically and theoretically. In terms of a block variant of the restricted isometry property of measurement matrix, we present weaker sufficient conditions for exact and robust block-sparse signal recovery than those known for l ₂/l ₁ minimization. We also propose an efficient Iteratively Reweighted Least-Squares (IRLS) algorithm for the induced non-convex optimization problem. The obtained weaker conditions and the proposed IRLS algorithm are tested and compared with the mixed l ₂/l ₁ minimization method and the standard l _q minimization method on a series of noiseless and noisy block-sparse signals. All the comparisons demonstrate the outperformance of the mixed l ₂/l _q (0 < q < 1) method for block-sparse signal recovery applications, and meaningfulness in the development of new CS technology.