Motivated by the application of neural networks in super resolution microscopy, this paper considers super resolution as the mapping of trigonometric moments of a discrete measure on $[0,1)^d$ to its support and weights. We prove that this map satisfies a local Lipschitz property where we give explicit estimates for the Lipschitz constant depending on the dimension $d$ and the sampling effort. Moreover, this local Lipschitz estimate allows to conclude that super resolution with the Wasserstein distance as the metric on the parameter space is even globally Lipschitz continuous. As a byproduct, we improve an estimate for the smallest singular value of multivariate Vandermonde matrices having pairwise clustering nodes.
翻译:在超分辨率显微镜应用神经网络的推动下,本文将超分辨率视为对其支持和重量的 $[0,1,1美元] 上离散测量的三角测量时间的绘图。 我们证明,该地图符合当地Lipschitz 属性,我们根据尺寸和取样努力对Lipschitz 常数作出明确估计。此外,当地Lipschitz 估计可以得出这样的结论,即以瓦塞斯坦距离作为参数空间的测量标准,甚至在全球范围都是连续的。作为副产品,我们改进了多变量 Vandermonde 矩阵最小单值的估算,并配对组合节点。