[Paper Review] Numerical stability bounds for algebraic systems of Prony type and their accurate solution by decimation

Dmitry Batenkov|arXiv (Cornell University)|Sep 10, 2014

Image and Signal Denoising Methods45 references3 citations

TL;DR

This paper provides component-wise asymptotic stability bounds for high-order Prony systems, establishing an absolute resolution limit for any method. It introduces a decimation technique—selecting equations via arithmetic progressions—to solve overdetermined Prony systems accurately, achieving near-optimal super-resolution without sacrificing numerical accuracy.

ABSTRACT

The Prony system of equations and its higher-order (confluent) generalizations appear prominently in many theoretical and applied problems. For instance, in signal processing these systems arise in recovery of sums of Diracs from a finite number of their Fourier measurements. The accurate and robust numerical solution of Prony type systems is considered to be a challenging problem, in particular reconstructing closely spaced nonlinear parameters (``nodes'', e.g. the support of the Diracs) in the presence of perturbed data. Our first contribution is providing component-wise asymptotic estimates for the numerical condition of the high-order Prony system, when the number of equations can in general be greater than the number of unknowns. These results provide, in particular, an absolute resolution limit for any method whatsoever. Our second contribution is proposing a technique for the overdetermined Prony problem with closely spaced nodes by ``decimation'', i.e. taking subsets of the equations with indices belonging to arithmetic progressions, and subsequently solving the resulting square systems. We show that solution of a decimated system is as accurate as the solution to the full overdetermined problem. Thus, decimation provides a tool to achieve near-optimal super-resolution.

Motivation & Objective

• To derive asymptotic component-wise condition estimates for high-order Prony systems, especially in overdetermined cases with more equations than unknowns.
• To establish an absolute resolution limit for any numerical method solving Prony-type systems.
• To develop a robust numerical technique for solving overdetermined Prony systems with closely spaced nodes under noisy data.
• To demonstrate that decimation—selecting equations via arithmetic progressions—preserves solution accuracy compared to solving the full overdetermined system.

Proposed method

• Deriving component-wise asymptotic bounds on the numerical condition number of high-order Prony systems using perturbation analysis.
• Introducing a decimation strategy that selects a subset of equations based on arithmetic progressions of indices to form square subsystems.
• Solving the resulting square Prony systems using standard numerical solvers, leveraging the structure of the decimated system to maintain accuracy.
• Proving that the solution of a decimated system is as accurate as the solution to the full overdetermined system, under mild assumptions on node separation.
• Analyzing the stability and resolution limits of the decimated system in relation to the condition number and node clustering.
• Using confluent Prony systems to model sums of Diracs with closely spaced supports, relevant to super-resolution in signal processing.

Experimental results

Research questions

• RQ1 What is the fundamental numerical stability limit for solving high-order Prony systems with closely spaced nodes?
• RQ2 Can a subset of equations from an overdetermined Prony system yield a solution as accurate as solving the full system?
• RQ3 How does decimation via arithmetic progression selection affect the resolution and accuracy of Prony system solutions?
• RQ4 What is the theoretical resolution limit for any method solving Prony-type systems, and how is it derived from condition number estimates?
• RQ5 Under what conditions does decimation preserve the accuracy of the full overdetermined system solution?

Key findings

• The paper establishes component-wise asymptotic bounds on the condition number of high-order Prony systems, providing a theoretical resolution limit for any numerical method.
• The decimation technique—selecting equations via arithmetic progressions—yields solutions as accurate as those from the full overdetermined system.
• The method achieves near-optimal super-resolution by effectively mitigating ill-conditioning in systems with closely spaced nodes.
• The accuracy of the decimated system is preserved even when the original system is overdetermined and ill-conditioned due to node clustering.
• The theoretical resolution limit derived from condition number estimates applies universally to all methods solving Prony-type systems.
• The decimation approach enables robust and accurate recovery of nonlinear parameters (e.g. Dirac supports) from noisy Fourier measurements.