On Overfitting In Discrete Super-Resolution Recovery

This paper studies the overfitting in discrete super-resolution problem. In particular, we solve for the estimate that simply overfits the noisy measurements. By doing this, we no longer require the prior knowledge of additive noise to set the parameter of sparse reconstruction algorithm to ensure the feasibility of target signal. The analysis of overfitting is based on a new proof of the quotient property of deterministic Fourier measurement matrix as well as a novel insight of the widely used interpolation-based proof technique in super-resolution literature. Our theoretical result shows that a similar stability guarantee holds for the overfitting algorithm as those for non-overfitting ones. The derived error bound is demonstrated by the numerical experiments.