Super-resolution limit of the ESPRIT algorithm
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The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its M+1 consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is 1/M and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than 1/M apart, this recovery problem is highly challenging and many existing algorithms either cannot deal with this situation or require restrictive assumptions on the sign of the measure. ESPRIT (Estimation of Signal Parameters via Rotation Invariance Techniques) is an efficient method that does not depend on the sign of the measure and this paper provides a stability analysis: we prove an explicit upper bound on the recovery error of ESPRIT in terms of the minimum singular value of Vandermonde matrices. Using prior estimates for the minimum singular value explains its resolution limit -- when the support of μ consists of multiple well-separated clumps, the noise level that ESPRIT can tolerate scales like SRF^-(2λ -2), where the super-resolution factor SRF governs the difficulty of the problem and λ is the cardinality of the largest clump. Our theory is validated by numerical experiments.
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