Robust off-grid recovery from compressed measurements

In this paper, the robust off-grid recovery of the compressed signals with atomic norm-regularized least-squares problem is studied. The aim of the recovery is to reconstruct the original signal and to detect its off-grid support set. The general optimality conditions for the solution to this problem and its dual problem are proposed and discussed. A method based on dual certification to detect the support set is introduced and proved to be effective. As a specific case, the target signal is further assumed to have unknown line spectrum. Then the problem is also an estimation of a low dimensional subspace which is indexed by continuous parameters, yet the dimension itself is unknown. Under these presumptions, the squared-error of the reconstruction is derived. Finally, numerical experiments are demonstrated in such case to validate the effectiveness of the method and the plausibility of the theory.