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An Optimal Recovery Condition for Sparse Signals with Partial Support Information via OMP

  • Huanmin Ge
  • Wengu ChenEmail author
Article
  • 35 Downloads

Abstract

This paper considers the orthogonal matching pursuit (OMP) algorithm for sparse recovery in both noiseless and noisy cases when the partial prior information is available. The prior information is included in an estimated subset of the support of the sparse signal. First, we show that if \(\varvec{A}\) satisfies \(\delta _{k+b+1}<\frac{1}{\sqrt{k-g+1}}\), then the OMP algorithm can perfectly recover any k-sparse signal \(\varvec{x}\) from \(\varvec{y}=\varvec{Ax}\) in \(k-g\) iterations when the prior support of \(\varvec{x}\) includes g true indices and b wrong indices. Furthermore, we show that the condition \(\delta _{k+b+1}<\frac{1}{\sqrt{k-g+1}}\) is optimal. Second, we achieve the exact recovery of the remainder support (i.e., it is composed of indices in the true support of \(\varvec{x}\) but not in the prior support) from \(\varvec{y}=\varvec{Ax}+\varvec{v}\) under appropriate conditions. On the other hand, for the remainder support recovery, we also obtain a necessary condition based on the minimum magnitude of nonzero elements in the remainder support of \(\varvec{x}\). Compared to the OMP algorithm, numerical experiments demonstrate that the OMP algorithm with the partial prior information has better recovery performance.

Keywords

Orthogonal matching pursuit Partial support information Restricted isometry constant Sensing matrix 

Notes

Acknowledgements

The authors thank the referees for their valuable suggestion and comments that greatly improve the presentation of this paper. The authors are also thankful to Qun Mo for his meaningful discussion about Remark 3. This work was supported by the NSF of China (Nos. 11871109, 61471343), the National Key Research and Development Program of China (No. 2018YFC2000605), NSAF(Grant No. U1830107) and the Science Challenge Project (Grant TZ2018001).

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.The Sports Engineering CollegeBeijing Sport UniversityBeijingChina
  2. 2.Institute of Applied Physics and Computational MathematicsBeijingChina

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