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Computable performance guarantees for compressed sensing matrices.

Myung Cho1, Kumar Vijay Mishra1, Weiyu Xu1

  • 1Department of ECE, University of Iowa, Iowa City, 52242 IA USA.

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|March 6, 2018
PubMed
Summary
This summary is machine-generated.

Verifying the null space condition for sparse signal recovery via ℓ1 minimization is challenging. This study introduces polynomial-time algorithms and a tree search method that precisely and quickly verify this condition, outperforming prior techniques.

Keywords:
Compressed sensingNull space conditionPerformance guaranteeSensing matrixℓ1 minimization

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Area of Science:

  • Signal Processing
  • Compressed Sensing
  • Optimization Theory

Background:

  • The null space condition is crucial for unique sparse signal recovery using ℓ1 minimization in compressed sensing.
  • Verifying this condition is computationally intensive, with existing methods providing only bounds on key parameters.

Purpose of the Study:

  • To develop efficient algorithms for establishing upper bounds on the null space condition's proportion parameter.
  • To introduce a precise and fast method for verifying the null space condition.

Main Methods:

  • Development of new polynomial-time algorithms to determine upper bounds for the proportion parameter.
  • Implementation of a novel tree search algorithm for accurate and rapid verification of the null space condition.

Main Results:

  • The proposed algorithms efficiently establish upper bounds for the proportion parameter.
  • The tree search algorithm demonstrates superior speed and accuracy in verifying the null space condition compared to existing methods.
  • Numerical experiments confirm the enhanced performance over linear programming (LP) and semidefinite programming (SDP) approaches.

Conclusions:

  • The developed polynomial-time algorithms and tree search method offer a significant advancement in verifying the null space condition for compressed sensing.
  • These new techniques provide a computationally efficient and accurate solution for assessing the unique recoverability of sparse signals.