Towards a Theory of Stable Super-Resolution: Model-Based Formulation and Stability Analysis

Abstract

In mathematics, a super-resolution problem can be formulated as acquiring high-frequency data from low-frequency measurements. This extrapolation problem in the frequency domain is well-known to be unstable. We propose a model-based super-resolution framework (Model-SR) for solving the super-resolution problem and analyzing its stability, aiming to narrow the gap between limited theory and the broad empirical success of super-resolution methods. The key rationale is that, to be determined by its low-frequency components, the target signal must possess a low-dimensional structure. Instead of assuming that the signal itself lies on a low-dimensional manifold in the signal space, we assume that it is generated from a model with a low-dimensional parameter space. This shift of perspective allows us to analyze stability directly through the model parameters. Within this framework, we can recover the signal by solving a nonlinear least square problem and achieve super-resolution by extracting its high-frequency components. Theoretically, the resolution-enhancing map is proven to have Lipschitz continuity, with a constant that depends crucially on parameter separation conditions\. This separation condition can be effectively enforced via sparsity modeling, which requires using the minimal number of parameters to represent the measured signal, thereby highlighting the role of sparsity in the stability of super-resolution. Moreover, the Lipschitz constant grows with the high-frequency cutoff, ultimately rendering extrapolation ineffective beyond a certain threshold. We apply the general theory to three concrete models and give the stability estimates for each model. Numerical experiments are conducted to show the super-resolution behavior of the proposed framework. The model-based mathematical framework can be extended to problems with similar structures.