The Generalized Operator Based Prony Method

Abstract

The generalized Prony method introduced in [13] is a reconstruction technique for a large variety of sparse signal models that can be represented as sparse expansions into eigenfunctions of a linear operator A. However, this procedure requires the evaluation of higher powers of the linear operator A that are often expensive to provide.In this paper we propose two important extensions of the generalized Prony method that simplify the acquisition of the needed samples essentially and at the same time can improve the numerical stability of the method. The first extension regards the change of operators from A to ϕ(A), where ϕ is an analytic function, while A and ϕ(A) possess the same set of eigenfunctions. The goal is now to choose ϕ such that the powers of ϕ(A) are much simpler to evaluate than the powers of A. The second extension concerns the choice of the sampling functionals. We show, how new sets of different sampling functionals F k can be applied with the goal to reduce the needed number of powers of the operator A (resp. ϕ(A)) in the sampling scheme and to simplify the acquisition process for the recovery method.

Learning algebraic decompositions using Prony structures

“…Several applications for various choices of the endomorphism ϕ and the functional ∆ can be found in [22], for example, with ϕ ∈ End(W ) chosen as a Sturm-Liouville differential operator (W = C ∞ (R)) or as a diagonal matrix with distinct elements on the diagonal (W = K n ). [21] and Stampfer-Plonka [31]. At present Prony structures do not cover this variation.…”

“…Classic applications of Prony's method include for example Sylvester's method for Waring decompositions of binary forms [32,33] and Padé approximation [36]. Since then these tools have been further developed [26,22,27,31] and recently also advances have been made on multivariate versions. Direct attempts can be found in, e.g., [28,23,18,17,30,16,21], for methods based on projections to univariate exponential sums see, e.g., [7,8,6].…”

Super-resolution for doubly-dispersive channel estimation

“…Finally, we like to mention that there exist other methods for super-resolution as Prony-like methods [13,30,31,37,38]. These are spectral methods which perform spike localization from low frequency measurements.…”

Modifications of Prony's Method for the Reconstruction of Structured Functions

“…Furthermore, this theory also fits into the framework of the generalized operator based Prony method, see [Sta18,SP20] and can be interpreted as a changing of operators, see [SP20], Theorem 3.4.…”

“…Most notably are the generalized Prony method derived by Peter and Plonka in 2013, see [PP13], as well as the generalized operator based Prony method developed by Plonka and Stampfer in 2020, see [Sta18,SP20]. We will give a brief summary of the generalized Prony method as described in [PP13], and begin with reformulating the classical Prony method.…”