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.
翻译:根据光谱分散的信号$mathbf{y} =\ sum ⁇ i= 1 =1 ⁇ s x_ i\ mathbf{f} (\ tau_i)\ i)\ 美元, 由 $s 构成的光谱分散的信号$\ mathbf{c{y} = = = SSM 类超分辨率算法比基于 ef- definite 编程的其他方法更有效。 我们的分析表明, 由 OMP 初始化成 OMP 的两阶段算法可以在 $n\ D @ 2n+1 包含 美元复杂的类内恢复频率组件, 我们考虑超分辨率问题, 是关于估算 $\ textx} (\ tau_ i_ i= dyn} (\ mathbf{x} $\ x) 美元。 我们进一步显示, 香草L OMP 算算算成的Slip- OMP 运算法, 一个在 OMP\\\ rxxxxxxxxxxxxxxxx 中, 一个可进一步的变换。