Super-Resolution Limit of the ESPRIT Algorithm

The problem of imaging point objects can be formulated as estimation of an unknown atomic measure from its <inline-formula> <tex-math notation="LaTeX">${M}+1$ </tex-math></inline-formula> consecutive noisy Fourier coefficients. The standard resolution of this inverse problem is <inline-formula> <tex-math notation="LaTeX">$1/{M}$ </tex-math></inline-formula> and super-resolution refers to the capability of resolving atoms at a higher resolution. When any two atoms are less than <inline-formula> <tex-math notation="LaTeX">$1/{M}$ </tex-math></inline-formula> 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 is an efficient method which does not depend on the sign of the measure. This paper provides an explicit error bound on the support matching distance of ESPRIT in terms of the minimum singular value of Vandermonde matrices. When the support consists of multiple well-separated clumps and noise is sufficiently small, the support error by ESPRIT scales like <inline-formula> <tex-math notation="LaTeX">$\text {SRF}^{2\lambda -2} \times \text {Noise}$ </tex-math></inline-formula>, where the Super-Resolution Factor (SRF) governs the difficulty of the problem and <inline-formula> <tex-math notation="LaTeX">$\lambda $ </tex-math></inline-formula> is the cardinality of the largest clump. Our error bound matches the min-max rate of a special model with one clump of closely spaced atoms up to a factor of <inline-formula> <tex-math notation="LaTeX">$M$ </tex-math></inline-formula> in the small noise regime, and therefore establishes the near-optimality of ESPRIT. Our theory is validated by numerical experiments.