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

We consider the inverse problem of recovering locations and amplitudes a collection point sources represented as discrete measure, given M+1 its noisy low-frequency Fourier coefficients. Super-resolution refers to stable recovery when distance Δ between two closest is less than 1/M. introduce clumps model where are closely spaced within several clumps. Under this assumption, we derive non-asymptotic lower bound for minimum singular value Vandermonde matrix whose nodes determined by sources. Our estimate weighted ℓ2 sum, each term only depends on configuration individual clump. The main novelty that our obtains an exact dependence Super-Resolution Factor SRF=(MΔ)−1. As noise level increases, sensitivity noise-space correlation function in MUSIC algorithm degrades according power law SRF exponent cardinality largest Numerical experiments validate theoretical bounds MUSIC. also provide upper min-max error super-resolution grid model, which turn related matrices.