On the accuracy of Prony's method for recovery of exponential sums with closely spaced exponents

TL;DR

This paper analyzes the accuracy and stability of Prony's method for super-resolution, showing it is optimal and numerically stable even when the support points are very closely spaced, surpassing previous bounds.

Contribution

It provides the first detailed accuracy bounds for Prony's method in super-resolution with minimal separation below the Rayleigh limit, including stability analysis.

Findings

Prony's method is optimal under certain conditions.

Prony's method remains numerically stable in finite-precision arithmetic.

The analysis reveals cancellations that improve understanding of error propagation.

Abstract

In this paper we establish accuracy bounds of Prony's method (PM) for recovery of sparse measures from incomplete and noisy frequency measurements, or the so-called problem of super-resolution, when the minimal separation between the points in the support of the measure may be much smaller than the Rayleigh limit. In particular, we show that PM is optimal with respect to the previously established min-max bound for the problem, in the setting when the measurement bandwidth is constant, with the minimal separation going to zero. Our main technical contribution is an accurate analysis of the inter-relations between the different errors in each step of PM, resulting in previously unnoticed cancellations. We also prove that PM is numerically stable in finite-precision arithmetic. We believe our analysis will pave the way to providing accurate analysis of known algorithms for the super-resolution problem in full generality.