论文标题
超分辨率OMP的性能分析
Performance Analysis of OMP in Super-Resolution
论文作者
论文摘要
Given a spectrally sparse signal $\mathbf{y} = \sum_{i=1}^s x_i\mathbf{f}(τ_i) \in \mathbb{C}^{2n+1}$ consisting of $s$ complex sinusoids, we consider the super-resolution problem, which is about estimating frequency components $ \ {τ_i\} _ {i = 1}^s $ $ \ mathbf y $。我们考虑用于超分辨率的OMP型算法，该算法比基于半明确编程的其他方法更有效。我们的分析表明，具有OMP初始化的两阶段算法可以在分离条件下恢复频率组件$nδ\ gtrsim \ text {dyn}（\ Mathbf {x}）$，并且对$ \ text {dyn}}（\ naterbf {x}）的依赖是不可避免的。我们进一步表明，在每次迭代时具有附加的完善步骤的Slid-Omp-tomp算法是在分离条件$nδ\ geq c $下恢复$ \ {τ_i\} _ = 1}^s $。此外，我们的结果可以扩展到使用$ O（S^2 \ log n）$测量的不完整测量模型。
Given a spectrally sparse signal $\mathbf{y} = \sum_{i=1}^s x_i\mathbf{f}(τ_i) \in \mathbb{C}^{2n+1}$ consisting of $s$ complex sinusoids, we consider the super-resolution problem, which is about estimating frequency components $\{τ_i\}_{i=1}^s$ 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 $nΔ\gtrsim \text{dyn}(\mathbf{x})$ 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 $\{τ_i\}_{i=1}^s$ under the separation condition $nΔ\geq c$. Moreover, our result can be extended to an incomplete measurement model with $O( s^2\log n)$ measurements.
