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We consider the problem of finding a sparse solution for an underdeterminedlinear system of equations when the known parameters on both sides of thesystem are subject to perturbation. This problem is particularly relevant toreconstruction in fully-perturbed compressive-sensing setups where both theprojected measurements of an unknown sparse vector and the knowledge of theassociated projection matrix are perturbed due to noise, error, mismatch, etc.We propose a new iterative algorithm for tackling this problem. The proposedalgorithm utilizes the proximal-gradient method to find a sparse totalleast-squares solution by minimizing an l1-regularized Rayleigh-quotient costfunction. We determine the step-size of the algorithm at each iteration usingan adaptive rule accompanied by backtracking line search to improve thealgorithm's convergence speed and preserve its stability. The proposedalgorithm is considerably faster than a popular previously-proposed algorithm,which employs the alternating-direction method and coordinate-descentiterations, as it requires significantly fewer computations to deliver the sameaccuracy. We demonstrate the effectiveness of the proposed algorithm viasimulation results.

Fast Reconstruction Algorithm for Perturbed Compressive Sensing Based on Total Least-Squares and Proximal Splitting