关于采用Prony方法恢复指数紧邻的指数和的精度研究
On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents
AI总结：
本研究针对超分辨率问题中测度支撑点间距小于瑞利极限的场景，分析Prony方法的误差抵消效应，证明其在有限精度下的数值稳定性及相对极小极大界的最优性，为超分辨率算法精度分析提供基础。
摘要：
本文针对稀疏测度支撑点的最小间距可能远小于瑞利极限的情况，建立了Prony方法（PM）从不完整且含噪的频率测量中恢复稀疏测度（即所谓的超分辨率问题）的精度界。具体而言，我们证明在测量带宽恒定、最小间距趋于零的设定下，PM相对于该问题此前建立的极小极大界是最优的。我们的主要技术贡献是对PM各步骤中不同误差之间的相互关系进行了精确分析，发现了此前未被注意到的误差抵消效应。我们还证明了PM在有限精度算术下具有数值稳定性。我们相信，本研究的分析将为全面分析超分辨率问题现有算法的精度奠定基础。
英文摘要：
In this paper we establish accuracy bounds of Prony's method (PM) for recovery of sparse measures from incomplete and noisy frequency measurements, or the so-called problem of super-resolution, when the minimal separation between the points in the support of the measure may be much smaller than the Rayleigh limit. In particular, we show that PM is optimal with respect to the previously established min-max bound for the problem, in the setting when the measurement bandwidth is constant, with the minimal separation going to zero. Our main technical contribution is an accurate analysis of the inter-relations between the different errors in each step of PM, resulting in previously unnoticed cancellations. We also prove that PM is numerically stable in finite-precision arithmetic. We believe our analysis will pave the way to providing accurate analysis of known algorithms for the super-resolution problem in full generality.
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