### Abstract

If a signal x is known to have a sparse representation with respect to a frame, the signal can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. This paper analyzes the mean squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal. The first main result is an MSE bound that depends on a new bound on approximating a Gaussian signal as a linear combination of elements of an overcomplete dictionary. This bound may be of independent interest for source coding. Further analyses are for dictionaries generated randomly according to a spherically-symmetric distribution and signals expressible with single dictionary elements. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. In the limit of large dimension, these approximations have simple forms. The asymptotic expressions reveal a critical input signal-to-noise ratio (SNR) for signal recovery.

Original language | English (US) |
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Title of host publication | Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05 |

Pages | 1706-1710 |

Number of pages | 5 |

Volume | 2005 |

DOIs | |

State | Published - 2005 |

Event | 2005 IEEE International Symposium on Information Theory, ISIT 05 - Adelaide, Australia Duration: Sep 4 2005 → Sep 9 2005 |

### Other

Other | 2005 IEEE International Symposium on Information Theory, ISIT 05 |
---|---|

Country | Australia |

City | Adelaide |

Period | 9/4/05 → 9/9/05 |

### Fingerprint

### ASJC Scopus subject areas

- Electrical and Electronic Engineering

### Cite this

*Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05*(Vol. 2005, pp. 1706-1710). [1523636] https://doi.org/10.1109/ISIT.2005.1523636

**Analysis of denoising by sparse approximation with random frame asymptotics.** / Fletcher, Alyson K.; Rangan, Sundeep; Goyal, Vivek K.; Ramchandran, Kannan.

Research output: Chapter in Book/Report/Conference proceeding › Conference contribution

*Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05.*vol. 2005, 1523636, pp. 1706-1710, 2005 IEEE International Symposium on Information Theory, ISIT 05, Adelaide, Australia, 9/4/05. https://doi.org/10.1109/ISIT.2005.1523636

}

TY - GEN

T1 - Analysis of denoising by sparse approximation with random frame asymptotics

AU - Fletcher, Alyson K.

AU - Rangan, Sundeep

AU - Goyal, Vivek K.

AU - Ramchandran, Kannan

PY - 2005

Y1 - 2005

N2 - If a signal x is known to have a sparse representation with respect to a frame, the signal can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. This paper analyzes the mean squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal. The first main result is an MSE bound that depends on a new bound on approximating a Gaussian signal as a linear combination of elements of an overcomplete dictionary. This bound may be of independent interest for source coding. Further analyses are for dictionaries generated randomly according to a spherically-symmetric distribution and signals expressible with single dictionary elements. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. In the limit of large dimension, these approximations have simple forms. The asymptotic expressions reveal a critical input signal-to-noise ratio (SNR) for signal recovery.

AB - If a signal x is known to have a sparse representation with respect to a frame, the signal can be estimated from a noise-corrupted observation y by finding the best sparse approximation to y. This paper analyzes the mean squared error (MSE) of this denoising scheme and the probability that the estimate has the same sparsity pattern as the original signal. The first main result is an MSE bound that depends on a new bound on approximating a Gaussian signal as a linear combination of elements of an overcomplete dictionary. This bound may be of independent interest for source coding. Further analyses are for dictionaries generated randomly according to a spherically-symmetric distribution and signals expressible with single dictionary elements. Easily-computed approximations for the probability of selecting the correct dictionary element and the MSE are given. In the limit of large dimension, these approximations have simple forms. The asymptotic expressions reveal a critical input signal-to-noise ratio (SNR) for signal recovery.

UR - http://www.scopus.com/inward/record.url?scp=33749446056&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=33749446056&partnerID=8YFLogxK

U2 - 10.1109/ISIT.2005.1523636

DO - 10.1109/ISIT.2005.1523636

M3 - Conference contribution

AN - SCOPUS:33749446056

SN - 0780391519

SN - 9780780391512

VL - 2005

SP - 1706

EP - 1710

BT - Proceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05

ER -