The paper analyzes the accuracy of Prony's method (PM) for recovering exponential sums from incomplete and noisy frequency measurements, in the context of the super-resolution (SR) problem. The key contributions are:
Establishing that PM is optimal with respect to the previously derived min-max bounds for the SR problem, in the setting when the measurement bandwidth is constant and the minimal separation between the exponents goes to zero.
Providing a detailed error analysis of the different steps of PM, revealing previously unnoticed cancellations between the errors. This contrasts with a "naive" analysis, which leads to overly pessimistic bounds.
Proving that PM is numerically stable in finite-precision arithmetic.
The analysis focuses on the case where the exponents form a clustered configuration, with the largest cluster having size ℓ*. The authors show that for constant bandwidth Ω and minimal separation δ → 0, the node errors scale as δ^(2-2ℓ*) ϵ and the amplitude errors scale as δ^(1-2ℓ*) ϵ for ℓ* > 1, and ϵ for ℓ* = 1. These bounds match the previously established min-max limits.
The authors believe their analysis paves the way for providing accurate analysis of other high-resolution algorithms for the super-resolution problem.
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