### 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 |

DOIs | |

State | Published - Dec 1 2005 |

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

### Publication series

Name | IEEE International Symposium on Information Theory - Proceedings |
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Volume | 2005 |

ISSN (Print) | 2157-8099 |

### Other

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

City | Adelaide |

Period | 9/4/05 → 9/9/05 |

### Fingerprint

### ASJC Scopus subject areas

- Theoretical Computer Science
- Information Systems
- Modeling and Simulation
- Applied Mathematics

### Cite this

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