Exact Reconstruction using Beurling Minimal Extrapolation (1103.4951v2)

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.

• The paper introduces the Generalized Minimal Extrapolation (GME) algorithm for the exact reconstruction of signed discrete measures from a finite number of generalized moments.
• It proves that nonnegative measures with finite support are uniquely recoverable from just 2s+1 generalized moments, where s is the support size.
• The paper proposes constructing deterministic sensing matrices for compressed sensing of nonnegative signals, reducing sampling requirements independent of ambient dimension.

Exact Reconstruction using Beurling Minimal Extrapolation: A Summary

The research paper titled "Exact Reconstruction using Beurling Minimal Extrapolation" authored by Yohann de Castro and Fabrice Gamboa explores the reconstruction of measures with finite support using an innovative algorithm named Generalized Minimal Extrapolation (GME). This work extends the principles of compressed sensing to the domain of signed measures, proposing mechanisms to achieve exact reconstruction from a finite set of generalized moments.

Overview of Key Concepts

• Generalized Minimal Extrapolation (GME): GME is an algorithm designed to reconstruct measures accurately using a finite number of generalized moments, which include standard moments, Laplace transformation, and Stieltjes transformation. It minimizes the total variation, analogous to the $\ell_1$-norm minimization in basis pursuit used in compressed sensing.
• Measure Framework: The paper considers signed discrete measures over a specified real interval, typically $I = [-1, 1]$. Here, the measures are expressed in terms of Dirac measures with unknown supports and weights.
• Extrema Jordan Type Measures: A novel framework is introduced for categorizing measures whose supports and weights can be uniquely determined through total variation minimization. Extrema Jordan type measures are defined such that their Jordan decomposition (the representation of a measure as a difference of two non-negative measures) satisfies specific support and interpolation constraints.
• Generalized Dual Polynomials: These polynomials serve as an interpolation tool, aiding the reconstruction by targeting specific supports in the measure space. The existence of such polynomials is a sufficient condition for the unique recovery via GME.

Numerical Results and Implications

1. Exact Reconstruction: The paper proves that nonnegative measures with finite support are uniquely recoverable from $2s+1$ generalized moments, where $s$ is the number of points in the support. This is particularly significant as it offers deterministic solutions akin to randomized strategies in compressed sensing.
2. Deterministic Sensing Matrices: The authors propose construction methodologies for deterministic matrices in compressed sensing contexts for nonnegative signals, lowering the bound for sampling requirements. This finding does not scale with the ambient dimension $p$, offering efficient solutions for sparse signal reconstruction.
3. Generalized Chebyshev Measures: By employing Chebyshev polynomials in the generalized setting, the paper offers reconstruction frameworks for structured measures, further broadening the applicability of GME.
4. Nullspace Property: An adaptation of the nullspace property, fundamental to compressed sensing, is presented. This concept verifies the feasibility and exact recovery within the Jordan support families defined for the measures in consideration.

Speculation on Future Developments

This exploration opens up several avenues for further investigation. Insights into extending GME for measures outside the real line, or adapting it to incorporate noise or perturbations in moment measurements, could significantly enhance its applicability. Additionally, drawing parallels with more comprehensive frameworks in AI could bridge gaps between theoretical perspectives and practical implementations.

Conclusion

The paper represents significant progress in bridging gaps between the measure theory and compressed sensing, presenting new techniques for exact measure reconstruction. By effectively using generalized moments and presenting deterministic frameworks, the authors contribute substantially to ensuring reliable recovery in sparse settings. These results could be pivotal in fields like signal processing, statistics, and computational mathematics, where precise reconstruction is of essence.