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

Stable super-resolution limit and smallest singular value of restricted Fourier matrices