Beyond ℓ1 norm minimization — High quality recovery of non-sparse compressible signals

We propose a novel algorithm for the recovery of non-sparse, but compressible signals from linear undersampled measurements. The algorithm proposed in this paper consists of two steps. The first step recovers the signal by the ℓ
1
minimization. Then, the second step decomposes the ℓ
1
reconstruction into major and minor components. By using the major components, measurements for the minor components of the target signal are estimated. Error evaluation of the estimate leads to the standard ridge regression for the recovery of the minor components with the regularization parameter determined using the error bound. After a slight modification to the major components, the final estimate is obtained by combining the two estimates. Computational cost of the proposed algorithm is mostly the same as the ℓ
1
minimization. Simulation results show the effectiveness of the proposed algorithm over not only ℓ
1
minimization but also the Lasso estimator.