# Preconditioning for Orthogonal Matching Pursuit with Noisy and Random Measurements: The Gaussian Case

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## Abstract

The success of orthogonal matching pursuit (OMP) in the sparse signal recovery heavily depends on its ability for correct support recovery. Based on a support recovery guarantee for OMP expressed in terms of the mutual coherence, and a result about the concentration of the extreme singular values of a Gaussian random matrix, this paper proposes a preconditioning method for increasing the recovery rate of OMP from random and noisy measurements. Compared to several existing preconditionings, the proposed method can reduce the mutual coherence with a proven high probability. Simultaneously, the proposed preconditioning can also succeed with a high probability in providing slight signal-to-noise ratio reduction, which is empirically shown to be less severe than that caused by a recently suggested technique for the noisy case. The simulations show the advantages of the proposed preconditioning over other currently relevant ones in terms of both the performance improvement for OMP, and computation time.

## Keywords

Orthogonal matching pursuit Noise Preconditioning Gaussian random matrices Mutual coherence## Notes

### Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Contacts 11131006, 41390450 and 91330204, in part by EU FP7-IRSES Project LIVCODE under Grant 295151, and in part by the National Basic Research Program of China under Contact 2013CB329404. We would like to thank Evaggelia Tsiligianni for providing the code implementing the preconditioning in his or her own work, and patiently discussing the details about the work. Special thanks are due to Karin Schnass for instructive discussions on her work. We would also like to thank the referees for numerous suggestions which helped to clarify the exposition and argumentation.

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