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Related Experiment Video

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Sparse Recovery Beyond Compressed Sensing: Separable Nonlinear Inverse Problems.

Brett Bernstein1, Sheng Liu2, Chrysa Papadaniil3

  • 1Courant Institute of Mathematical Sciences, New York University, New York, NY 10011 USA.

IEEE Transactions on Information Theory
|September 14, 2020
PubMed
Summary
This summary is machine-generated.

This study introduces a new theory for sparse recovery in deterministic settings, enabling accurate parameter estimation from nonlinear measurements. Convex programming effectively recovers parameters when they are sufficiently distinct, advancing data analysis in various scientific fields.

Keywords:
Sparse recoveryconvex programmingcorrelated measurementsdual certificatesincoherencenonlinear inverse problemssource localization

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Area of Science:

  • Data analysis
  • Inverse problems
  • Signal processing

Background:

  • Extracting information from nonlinear measurements presents a significant challenge.
  • Separable inverse problems involve data modeled as nonlinear functions of parameters.
  • Existing convex programming methods lack theoretical justification for deterministic settings.

Purpose of the Study:

  • To develop a theoretical framework for sparse recovery in deterministic settings.
  • To provide theoretical justification for using convex programming in separable nonlinear inverse problems.
  • To demonstrate the applicability of the theory in real-world scenarios.

Main Methods:

  • Reformulating separable nonlinear inverse problems as underdetermined sparse-recovery problems.
  • Applying convex programming techniques to solve these problems.
  • Developing a new theory for sparse recovery adapted to deterministic measurement operators.

Main Results:

  • Convex programming successfully recovers parameters if they are sufficiently distinct relative to the measurement operator's correlation structure.
  • The proposed theory provides a foundation for sparse recovery in deterministic scenarios.
  • Numerical experiments validate the approach for heat-source localization and electroencephalography data.

Conclusions:

  • A robust theory for sparse recovery in deterministic settings has been established.
  • Convex programming is a viable method for parameter estimation in separable nonlinear inverse problems.
  • The findings have implications for diverse fields including medical imaging and geophysics.