Analysis of denoising by sparse approximation with random frame asymptotics

Alyson K. Fletcher, Sundeep Rangan, Vivek K. Goyal, Kannan Ramchandran

Research output: Chapter in Book/Report/Conference proceedingConference contribution

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 languageEnglish (US)
Title of host publicationProceedings of the 2005 IEEE International Symposium on Information Theory, ISIT 05
Pages1706-1710
Number of pages5
DOIs
StatePublished - Dec 1 2005
Event2005 IEEE International Symposium on Information Theory, ISIT 05 - Adelaide, Australia
Duration: Sep 4 2005Sep 9 2005

Publication series

NameIEEE International Symposium on Information Theory - Proceedings
Volume2005
ISSN (Print)2157-8099

Other

Other2005 IEEE International Symposium on Information Theory, ISIT 05
CountryAustralia
CityAdelaide
Period9/4/059/9/05

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ASJC Scopus subject areas

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

Cite this

Fletcher, A. K., Rangan, S., Goyal, V. K., & Ramchandran, K. (2005). Analysis of denoising by sparse approximation with random frame asymptotics. In 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