Performance Analysis of OMP in Super-Resolution

TL;DR

This paper analyzes the performance of OMP algorithms for super-resolution, demonstrating conditions under which they can accurately recover frequency components efficiently, including variants with refinement steps and incomplete measurements.

Contribution

The paper provides theoretical guarantees for OMP-based algorithms in super-resolution, including a two-stage approach and a refined Sliding-OMP method, under specific separation conditions.

Findings

OMP with two-stage initialization can recover frequencies if separation condition is met.

Sliding-OMP with refinement step can recover frequencies under a milder separation condition.

The methods extend to incomplete measurements with O(s^2 log n) samples.

Abstract

Given a spectrally sparse signal $\mathbf{y} = \sum_{i=1}^s x_i\mathbf{f}(\tau_i) \in \mathbb{C}^{2n+1}$ consisting of $s$ complex sinusoids, we consider the super-resolution problem, which is about estimating frequency components $\{\tau_i\}_{i=1}^s$ of $\mathbf y$. We consider the OMP-type algorithms for super-resolution, which is more efficient than other approaches based on Semi-Definite Programming. Our analysis shows that a two-stage algorithm with OMP initialization can recover frequency components under the separation condition $n\Delta \gtrsim \text{dyn}(\mathbf{x})$ and the dependency on $\text{dyn}(\mathbf{x})$ is inevitable for the vanilla OMP algorithm. We further show that the Sliding-OMP algorithm, a variant of the OMP algorithm with an additional refinement step at each iteration, is provable to recover $\{\tau_i\}_{i=1}^s$ under the separation condition $n\Delta \geq c$. Moreover, our result can be extended to an incomplete measurement model with $O( s^2\log n)$ measurements.