Upper bounds on the error of sparse vector and low-rank matrix recovery

Suppose that a solution x˜ to an underdetermined linear system b=Ax is given. x˜ is approximately sparse meaning that it has a few large components compared to other small entries. However, the total number of nonzero components of x˜ is large enough to violate any condition for the uniqueness of the sparsest solution. On the other hand, if only the dominant components are considered, then it will satisfy the uniqueness conditions. One intuitively expects that x˜ should not be far from the true sparse solution x0. It was already shown that this intuition is the case by providing upper bounds on ∥x˜−x0∥ which are functions of the magnitudes of small components of x˜ but independent from x0. In this paper, we tighten one of the available bounds on ∥x˜−x0∥ and extend this result to the case that b is perturbed by noise. Additionally, we generalize the upper bounds to the low-rank matrix recovery problem.