We consider the inverse problem of recovering the locations and amplitudes of a collection of point sources represented as a discrete measure, given $M+1$ of its noisy low-frequency Fourier coefficients. Super-resolution refers to a stable recovery when the distance Δ 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 Super-Resolution Factor $SRF={(M\mathrm{\Delta })}^{-1}$. As noise level increases, the 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.

在给定的条件下，我们考虑了恢复点源集合的位置和幅度的逆问题，这些点源表示为离散量度 $\mathrm{中号}+\mathrm{1个}$嘈杂的低频傅立叶系数。超分辨率是指两个最近点源之间的距离Δ小于$\mathrm{1个}/\mathrm{中号}$。我们引入了一个团块模型，其中点源在几个团块之间紧密间隔。在此假设下，我们得出范德蒙德矩阵的最小奇异值的非渐近下界，该范德蒙德矩阵的节点由点源确定。我们的估计值是加权的${\ell }^2$总和，其中每个术语仅取决于每个单独簇的配置。主要的新颖之处在于我们的下限获得了对超分辨率因子的精确依赖 $\mathrm{小号}\mathrm{[R}F={（\mathrm{中号}\mathrm{\Delta }）}^{-\mathrm{1个}}$。随着噪声水平的提高，MUSIC算法中噪声空间相关函数的灵敏度会根据SRF中的幂定律而降低，其中幂指数取决于最大簇的基数。数值实验验证了我们对最小奇异值和MUSIC灵敏度的理论界限。我们还为网格模型的超分辨率的最小-最大误差提供了上下限，而上下误差又与范德蒙德矩阵的最小奇异值密切相关。