We show that the exact recovery of sparse perturbations on the coefficient matrix in overdetermined Least Squares problems is possible for a large class of perturbation structures. The well established theory of Compressed Sensing enables us to prove that if the perturbation structure is sufficiently incoherent, then exact or stable recovery can be achieved using linear programming. We derive sufficiency conditions for both exact and stable recovery using known results of ℓ 0/ℓ 1 equivalence. However the problem turns out to be more complicated than the usual setting used in various sparse reconstruction problems. We propose and solve an optimization criterion and its convex relaxation to recover the perturbation and the solution to the Least Squares problem simultaneously. Then we demonstrate with numerical examples that the proposed method is able to recover the perturbation and the unknown exactly with high probability. The performance of the proposed technique is compared in blind identification of sparse multipath channels. © 2011 IEEE.