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Analysis of Orthogonal Matching Pursuit Using the Restricted Isometry Property

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2 Author(s)
Davenport, M.A. ; Dept. of Stat., Stanford Univ., Stanford, CA, USA ; Wakin, M.B.

Orthogonal matching pursuit (OMP) is the canonical greedy algorithm for sparse approximation. In this paper we demonstrate that the restricted isometry property (RIP) can be used for a very straightforward analysis of OMP. Our main conclusion is that the RIP of order K+1 (with isometry constant δ <; [ 1/( 3√K)]) is sufficient for OMP to exactly recover any K-sparse signal. The analysis relies on simple and intuitive observations about OMP and matrices which satisfy the RIP. For restricted classes of K-sparse signals (those that are highly compressible), a relaxed bound on the isometry constant is also established. A deeper understanding of OMP may benefit the analysis of greedy algorithms in general. To demonstrate this, we also briefly revisit the analysis of the regularized OMP (ROMP) algorithm.

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Information Theory, IEEE Transactions on  (Volume:56 ,  Issue: 9 )