# Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit

@article{Tropp2007SignalRF, title={Signal Recovery From Random Measurements Via Orthogonal Matching Pursuit}, author={Joel A. Tropp and Anna C. Gilbert}, journal={IEEE Transactions on Information Theory}, year={2007}, volume={53}, pages={4655-4666} }

This paper demonstrates theoretically and empirically that a greedy algorithm called orthogonal matching pursuit (OMP) can reliably recover a signal with m nonzero entries in dimension d given O(m ln d) random linear measurements of that signal. [... ] Key Result In some settings, the OMP algorithm is faster and easier to implement, so it is an attractive alternative to BP for signal recovery problems. Expand

## 8,014 Citations

Sparse Signal Recovery via Optimized Orthogonal Matching Pursuit

- Computer Science, Engineering2009 2nd International Congress on Image and Signal Processing
- 2009

The OOMP algorithm improves the Orthogonal Matching Pursuit (OMP) algorithm via providing the projection onto the subspace generated by the selected measurements and minimizing the corresponding residual error at each iteration.

Uniform Uncertainty Principle and Signal Recovery viaÂ Regularized Orthogonal Matching Pursuit

- Computer ScienceFound. Comput. Math.
- 2009

This paper finds a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization.

Subspace pursuit embedded in Orthogonal Matching Pursuit

- Computer ScienceTENCON 2012 IEEE Region 10 Conference
- 2012

A modification in OMP is proposed, using the well known Subspace Pursuit (SP), to refine the subspace estimated by OMP at any iteration and hence boost the sparse signal recovery performance of OMP.

Exact Recovery of Sparse Signals via Orthogonal Matching Pursuit: How Many Iterations Do We Need?

- Computer Science
- 2012

This analysis shows that OMP can accurately recover all $K-sparse signals within $\lceil 2.8 K \rceil$ iterations when the measurement matrix satisfies a restricted isometry property (RIP).

Multiple Candidate Matching Pursuit

- Computer Science
- 2012

This paper multiple candidate matching pursuit (MuCaMP), which builds up candidate support set in every iteration and uses the minimum residual at last iteration using the restricted isometry property (RIP), derives the sufficient condition for MuCaMP to recover the sparse signal exactly.

Title Signal recovery from random measurements viaextended orthogonal matching pursuit

- Computer Science
- 2018

This work has proposed a scheme that brings the required number of measurements for OMP closer to BP, and extended the idea of OMPÎ± to illustrate another recovery scheme called OMPâˆž, which can achieve a close to 0-norm recovery without any knowledge of m like BP.

Wavelet based sparse image recovery via Orthogonal Matching Pursuit

- Computer Science2014 Recent Advances in Engineering and Computational Sciences (RAECS)
- 2014

It is demonstrated that if orthogonal matching pursuit is implemented in multi stages, it gives a faster recovery of an image with kth Sparsity level by taking k ln R measurements for a dimension R.

Orthogonal Matching Pursuit with Tikhonov and Landweber Regularization

- Computer Science, MathematicsArXiv
- 2019

An iterative approach allows for a hardware efficient implementation of the OMP, and enables real-world applications of compressed sensing, and provides a series of numerical examples that demonstrate a good performance, especially when the number of measurements is relatively small.

Exact Recovery of Sparse Signals Using Orthogonal Matching Pursuit: How Many Iterations Do We Need?

- Computer ScienceIEEE Transactions on Signal Processing
- 2016

This analysis shows that OMP can accurately recover all K-sparse signals within [2.8 K] iterations when the measurement matrix satisfies a restricted isometry property (RIP).

A New Bound on Approximate Support Recovery

- Computer ScienceArXiv
- 2020

This result offers an affirmative answer to the conjecture of [Wang, TSP 2015] that the error rate of support recovery via OMP has no dependence on the maximum element of the signal.

## References

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Uniform Uncertainty Principle and Signal Recovery viaÂ Regularized Orthogonal Matching Pursuit

- Computer ScienceFound. Comput. Math.
- 2009

This paper finds a simple regularized version of Orthogonal Matching Pursuit (ROMP) which has advantages of both approaches: the speed and transparency of OMP and the strong uniform guarantees of L1-minimization.

Random Sampling of Sparse Trigonometric Polynomials, II.Â OrthogonalÂ Matching Pursuit versusÂ Basis Pursuit

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It is indicated that usually Basis Pursuit is significantly slower than greedy algorithms, while the recovery rates are very similar, and theoretical results on the success probability of reconstruction via Thresholding and OMP for both a continuous and a discrete probability model for the sampling points.

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Greed is good: algorithmic results for sparse approximation

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This article presents new results on using a greedy algorithm, orthogonal matching pursuit (OMP), to solve the sparse approximation problem over redundant dictionaries and develops a sufficient condition under which OMP can identify atoms from an optimal approximation of a nonsparse signal.

Matching pursuits with time-frequency dictionaries

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The authors introduce an algorithm, called matching pursuit, that decomposes any signal into a linear expansion of waveforms that are selected from a redundant dictionary of functions. Theseâ€¦

Orthogonal matching pursuit: recursive function approximation with applications to wavelet decomposition

- Computer Science, MathematicsProceedings of 27th Asilomar Conference on Signals, Systems and Computers
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A modification to the matching pursuit algorithm of Mallat and Zhang (1992) that maintains full backward orthogonality of the residual at every step and thereby leads to improved convergence is proposed.

Gradient Projection for Sparse Reconstruction: Application to Compressed Sensing and Other Inverse Problems

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This paper proposes gradient projection algorithms for the bound-constrained quadratic programming (BCQP) formulation of these problems and test variants of this approach that select the line search parameters in different ways, including techniques based on the Barzilai-Borwein method.

Stability Results for Random Sampling of Sparse Trigonometric Polynomials

- Computer ScienceIEEE Transactions on Information Theory
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It is shown that recovery by a BP variant is stable under perturbation of the samples values by noise, and the stability result is extended to (nonsparse) trigonometric polynomials that can be well approximated by sparse ones.

Sparse Solution Of Underdetermined Linear Equations By Stagewise Orthogonal Matching Pursuit

- Computer Science
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It is shown that for systems with â€˜typicalâ€™/â€˜randomâ€™ Î¦, a good approximation to the sparsest solution is obtained by applying a fixed number of standard operations from linear algebra, and rigorously derive a conditioned Gaussian distribution for the matched filtering coefficients at each stage of the procedure.

Homotopy continuation for sparse signal representation

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This work describes a homotopy continuation-based algorithm to find and trace efficiently all solutions of basis pursuit as a function of the regularization parameter, and shows the effectiveness of this algorithm in accurately and efficiently generating entire solution paths for basis pursuit.