Exact Reconstruction using Beurling Minimal Extrapolation

TL;DR

This paper demonstrates that finite support measures can be exactly reconstructed from a limited set of generalized moments using a novel extrapolation algorithm, extending compressed sensing principles to measures.

Contribution

It introduces a generalized minimal extrapolation algorithm for measures, extending basis pursuit and compressed sensing results to the measure framework with new deterministic sensing matrices.

Findings

Finite support measures are uniquely recoverable from 2s+1 generalized moments.

The method extends basis pursuit and compressed sensing concepts to measures.

Provides a new deterministic construction for sensing matrices in compressed sensing.

Abstract

We show that measures with finite support on the real line are the unique solution to an algorithm, named generalized minimal extrapolation, involving only a finite number of generalized moments (which encompass the standard moments, the Laplace transform, the Stieltjes transformation, etc). Generalized minimal extrapolation shares related geometric properties with basis pursuit of Chen, Donoho and Saunders [CDS98]. Indeed we also extend some standard results of compressed sensing (the dual polynomial, the nullspace property) to the signed measure framework. We express exact reconstruction in terms of a simple interpolation problem. We prove that every nonnegative measure, supported by a set containing s points,can be exactly recovered from only 2s + 1 generalized moments. This result leads to a new construction of deterministic sensing matrices for compressed sensing.