### Abstract

A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from m=4k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n→∞. This work strengthens this result by showing that a lower number of measurements, m=2k log(n-k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies k
_{min}≤ k ≤ k
_{max} but is unknown, m=2k
_{max} log(n-k
_{min}) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling m=2k log(n-k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.

Original language | English (US) |
---|---|

Article number | 6086631 |

Pages (from-to) | 1010-1021 |

Number of pages | 12 |

Journal | IEEE Transactions on Signal Processing |

Volume | 60 |

Issue number | 3 |

DOIs | |

State | Published - Mar 2012 |

### Fingerprint

### Keywords

- Compressed sensing
- detection
- lasso
- orthogonal matching pursuit (OMP)
- random matrices
- sparse approximation
- sparsity
- subset selection

### ASJC Scopus subject areas

- Electrical and Electronic Engineering
- Signal Processing

### Cite this

*IEEE Transactions on Signal Processing*,

*60*(3), 1010-1021. [6086631]. https://doi.org/10.1109/TSP.2011.2176936

**Orthogonal matching pursuit : A Brownian motion analysis.** / Fletcher, Alyson K.; Rangan, Sundeep.

Research output: Contribution to journal › Article

*IEEE Transactions on Signal Processing*, vol. 60, no. 3, 6086631, pp. 1010-1021. https://doi.org/10.1109/TSP.2011.2176936

}

TY - JOUR

T1 - Orthogonal matching pursuit

T2 - A Brownian motion analysis

AU - Fletcher, Alyson K.

AU - Rangan, Sundeep

PY - 2012/3

Y1 - 2012/3

N2 - A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from m=4k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n→∞. This work strengthens this result by showing that a lower number of measurements, m=2k log(n-k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies k min≤ k ≤ k max but is unknown, m=2k max log(n-k min) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling m=2k log(n-k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.

AB - A well-known analysis of Tropp and Gilbert shows that orthogonal matching pursuit (OMP) can recover a k-sparse n-dimensional real vector from m=4k log(n) noise-free linear measurements obtained through a random Gaussian measurement matrix with a probability that approaches one as n→∞. This work strengthens this result by showing that a lower number of measurements, m=2k log(n-k), is in fact sufficient for asymptotic recovery. More generally, when the sparsity level satisfies k min≤ k ≤ k max but is unknown, m=2k max log(n-k min) measurements is sufficient. Furthermore, this number of measurements is also sufficient for detection of the sparsity pattern (support) of the vector with measurement errors provided the signal-to-noise ratio (SNR) scales to infinity. The scaling m=2k log(n-k) exactly matches the number of measurements required by the more complex lasso method for signal recovery with a similar SNR scaling.

KW - Compressed sensing

KW - detection

KW - lasso

KW - orthogonal matching pursuit (OMP)

KW - random matrices

KW - sparse approximation

KW - sparsity

KW - subset selection

UR - http://www.scopus.com/inward/record.url?scp=84857217156&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84857217156&partnerID=8YFLogxK

U2 - 10.1109/TSP.2011.2176936

DO - 10.1109/TSP.2011.2176936

M3 - Article

VL - 60

SP - 1010

EP - 1021

JO - IEEE Transactions on Signal Processing

JF - IEEE Transactions on Signal Processing

SN - 1053-587X

IS - 3

M1 - 6086631

ER -