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

TL;DR

This work analyzes Prony's method for recovering sparse exponential sums from noisy Fourier samples in the super-resolution regime with closely spaced exponents. It establishes that, for constant bandwidth ${\Omega}$ and minimal separation ${\delta}\to 0$, PM achieves the min-max SR bounds in both node and amplitude estimation, aided by a delicate error-cancellation mechanism across PM's steps. It further proves numerical stability in finite-precision arithmetic and introduces the Decimated Prony Method (DPM), showing it attains the SR bounds in key cases and potentially in general settings. Comprehensive numerical experiments validate the theoretical predictions, highlighting the role of node projection and inter-cluster interactions, and linking the analysis to broader high-resolution SR algorithms.

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.

Figures (5)

• Figure 1: Left: the condition number $\kappa(H_n)$ of the Hankel matrix scales as $\delta^{2-2\ell}$. Middle: both $\Delta\boldsymbol{\mathrm{q}}$ and $|x_j-\tilde{x}_j|$ scale as $\delta^{2-2\ell}\epsilon$, showing that the errors in the coefficients of the Prony polynomial are not independent. For comparison, choosing a random perturbation of $\boldsymbol{\mathrm{p}}$ with same magnitude as $\Delta\boldsymbol{\mathrm{q}}$ and computing the roots $y_j$ of the resulting polynomial, we observe that $|y_j-x_j| = O(\delta^{3-3\ell}\epsilon)$, as predicted by prop:prony-naive-analysis. Right: the amplitude errors committed by Prony's method (blue) and by replacing the recovered nodes with random perturbations (red). Here, in contrast, the bound of prop:ampl-naive is not attained. All computations are done in floating-point arithmetic, with $\epsilon=10^{-15}$.
• Figure 2: Classical Prony method - asymptotic optimality. For cluster nodes $j=1,2$, the node errors $\mathcal{E}_{x,j}=|\tilde{x}_j-x_j|$ (left) scale is like $\delta^{2-2\ell}$, while the amplitude errors $\mathcal{E}_{a,j}=|\tilde{\alpha}_j-\alpha_j|$ (middle) scale like $\delta^{1-2\ell}$. For the non-cluster node $j=3$, both errors are bounded by a constant. Right: backward errors of each step, as specified in def:backward-errors, are on the order of machine epsilon, implying numerical stability of PM according to thm:finite-precision. Here $\epsilon=10^{-15}$ in all experiments.
• Figure 3: Prony's method: accuracy of amplitude recovery where the nodes are projected (left) or non-projected (center) prior to recovering the amplitudes. Right panel: the corresponding normalized amplitude discrepancy function $\mathcal{V}_1/\epsilon$ (see text). The results are consistent with the estimates \ref{['eq:TildVVBdClus']} (projected) and Theorem \ref{['Thm:ImprovedClust']} (non-projected).
• Figure 4: Decimated Prony Method - asymptotic optimality. For cluster nodes $j=1,2$, the node amplification factors $\mathcal{K}_x$ (left) scale like $\textrm{SRF}^{2\ell-2}$, and for the non-cluster node ($j=3$) it is bounded by a constant. The amplitude error amplification factors $\mathcal{K}_a$ for the non-cluster node with no projection (middle) are bounded by a constant while with projection (right) they scale like $\textrm{SRF}^{\ell-1}$. For the cluster nodes, both amplitude error amplification factors scale like $\textrm{SRF}^{2\ell-1}$.
• Figure 5: Matrix Pencil's asymptotic optimality. For cluster nodes, the node amplification factors $\mathcal{K}_x$ (left) scale like $\textrm{SRF}^{2\ell-2}$, and for the non-cluster node (j=3) it is bounded by a constant. The amplitude error amplification factors $\mathcal{K}_a$ for the non-cluster node with no projection (middle) are bounded by a constant while with projection (right) they scale like $\textrm{SRF}^{\ell-1}$ for large enough $\textrm{SRF}$. For the cluster nodes, both amplitude error amplification factors scale like $\textrm{SRF}^{2\ell-1}$.

Theorems & Definitions (48)

• Remark 1
• Definition 1: Minimax rate
• Remark 2
• Definition 2: Clustered configuration
• Theorem 1: batenkov2021b
• Proposition 1
• Proposition 2
• proof : Proof of prop:prony-naive-analysis
• Proposition 3
• proof