A Sharp Condition for Exact Support Recovery With Orthogonal Matching Pursuit
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Bibliographic record
Abstract
Support recovery of sparse signals from noisy measurements with orthogonal matching pursuit (OMP) has been extensively studied. In this paper, we show that for any K-sparse signal x, if a sensing matrix A satisfies the restricted isometry property (RIP) with restricted isometry constant δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K+ 1</sub> <; 1/√K + 1, then under some constraints on the minimum magnitude of nonzero elements of x, OMP exactly recovers the support of x from its measurements y = Ax + v in K iterations, where v is a noise vector that is ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sub> or ℓ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">∞</sub> bounded. This sufficient condition is sharp in terms of δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K+ 1</sub> since for any given positive integer K and any 1/√K + 1 ≤ δ <; 1, there always exists a matrix A satisfying the RIP with δ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">K+ 1</sub> = δ for which OMP fails to recover a K-sparse signal x in K iterations. Also, our constraints on the minimum magnitude of nonzero elements of x are weaker than existing ones. Moreover, we propose worst case necessary conditions for the exact support recovery of x, characterized by the minimum magnitude of the nonzero elements of x.
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Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.001 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
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