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Iterative Hard Thresholding and Normalized Iterative Hard Thresholding

Introduction

This README provides an overview of the implementation of the Iterative Hard Thresholding (IHT) and Normalized Iterative Hard Thresholding (NIHT) algorithms for compressed sensing. These algorithms are used to reconstruct sparse vectors from underdetermined measurements.

Dependencies

The implementation requires the following LaTeX packages:

  • amssymb
  • amsthm
  • amsmath
  • amsfonts
  • xy with the curve option
  • fullpage
  • enumerate
  • graphicx
  • algorithm
  • algorithmic
  • color
  • biblatex with numeric style

LaTeX Definitions and Commands

The code includes various LaTeX definitions and commands for convenience. These include custom math symbols, matrices, vectors, theorem styles, examples, and exercises.

Algorithms

Iterative Hard Thresholding (IHT)

IHT is an iterative algorithm used for compressed sensing. It aims to reconstruct sparse vectors from underdetermined measurements. The main function IHT takes an input matrix A, measurements y, sparsity level k, and an error tolerance tol. The output is a k-sparse approximation x_hat of the target signal x. The algorithm continues until the residual norm is less than the specified tolerance.

Normalized Iterative Hard Thresholding (NIHT)

NIHT is an improved version of IHT. It also aims to reconstruct sparse vectors from underdetermined measurements. The main function NIHT takes similar inputs to IHT, but it employs normalized steps for better stability. The algorithm continues until the residual norm is less than the specified tolerance.

Main Result

The main theorem proves the convergence of the NIHT algorithm. It states that under certain conditions on the asymmetric restricted isometry constants of the matrix A, the NIHT algorithm converges to the original sparse vector x. The theorem also provides an upper bound on the reconstruction error.

License

This code is provided under the MIT License. Feel free to use and modify it for your projects.

Acknowledgments

Special thanks to the authors of the algorithms and the source papers for their contributions.

Contact

For any questions or inquiries, please contact [email protected].

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