广义尖峰恢复的稳定性与超分辨率
Stability and super-resolution of generalized spike recovery
AI总结：
研究在有限傅里叶样本中恢复线性组合的狄拉克δ函数及其导数的稳定性与超分辨率问题，提出基于样本降采样的正则化方案以提升恢复性能。
摘要：
我们考虑从有限个受噪声干扰的傅里叶样本中恢复线性组合的狄拉克δ函数和导数的问题。这是一类广义的尖峰恢复问题，近期受到广泛关注。我们分析了该问题在两种不同设置下的数值条件，取决于量$Nη$的量级，其中$N$是傅里叶样本的数量，$η$是广义尖峰之间的最小距离。在“良好条件” regime $Nη\gg1$，我们为对应的最小二乘问题的解的一阶扰动提供了上界。在近碰撞或“超分辨率” regime $Nη\to0$且单个簇的情况下，我们提出了一种基于样本降采样的自然正则化方案，本质上增加了$η$的分离度，并在实践中展示了该方案的有效性和近最优性。
英文摘要：
We consider the problem of recovering a linear combination of Dirac delta functions and derivatives from a finite number of Fourier samples corrupted by noise. This is a generalized version of the well-known spike recovery problem, which is receiving much attention recently. We analyze the numerical conditioning of this problem in two different settings depending on the order of magnitude of the quantity $Nη$, where $N$ is the number of Fourier samples and $η$ is the minimal distance between the generalized spikes. In the "well-conditioned" regime $Nη\gg1$, we provide upper bounds for first-order perturbation of the solution to the corresponding least-squares problem. In the near-colliding, or "super-resolution" regime $Nη\to0$ with a single cluster, we propose a natural regularization scheme based on decimating the samples \textendash{} essentially increasing the separation $η$ \textendash{} and demonstrate the effectiveness and near-optimality of this scheme in practice.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
