We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given $M$ of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance $\Delta$ between the two closest point sources is less than $1/M$. We introduce a clumps model where the point sources are closely spaced within several clumps. Under this assumption, we derive a non-asymptotic lower bound for the minimum singular value of a Vandermonde matrix whose nodes are determined by the point sources. Our estimate is given as a weighted $\ell^2$ sum, where each term only depends on the configuration of each individual clump. The main novelty is that our lower bound obtains an exact dependence on the {\it Super-Resolution Factor} $SRF=(M\Delta)^{-1}$. As noise level increases, the {\it sensitivity of the noise-space correlation function in the MUSIC algorithm} degrades according to a power law in $SRF$ where the exponent depends on the cardinality of the largest clump. Numerical experiments validate our theoretical bounds for the minimum singular value and the sensitivity of MUSIC. We also provide lower and upper bounds for a min-max error of super-resolution for the grid model, which in turn is closely related to the minimum singular value of Vandermonde matrices.

翻译：我们考虑的是恢复一个点源集的位置和振幅的反面问题,它代表的点源集合是一个离散的计量标准,以美元计其噪音低频Fleier系数。超级分辨率是指当两个最接近点源之间的距离不到1美元/M美元时,美元/Delta美元之间的距离低于1美元/M美元时的稳定恢复。我们引入了一个点源在几个宽块中间隔很近的星云模型。在这个假设下,我们得出一个非不方便的下限,用于由点源决定节点的Vandermonde矩阵最小单值。我们的估计是按一个加权的 $@ell_2$和总和,其中每个术语仅取决于每个单块的配置。主要新颖之处是,我们下限的基底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底底