An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit
An Improved RIP-Based Performance Guarantee for Sparse Signal Recovery via Orthogonal Matching Pursuit
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DOI：
10.1109/tit.2014.2338314
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发表时间：
2014-01
影响因子：
2.5
通讯作者：
Ling-Hua Chang;Jwo-Yuh Wu
中科院分区：
文献类型：
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作者：
Ling-Hua Chang;Jwo-Yuh Wu
最近报道的通过正交匹配追踪（OMP）在K次迭代（当没有噪声时）中完美恢复K稀疏向量的充分条件是感测矩阵的受限等距常数（RIC）满足δK+1 <;（1/RIC（K）+ 1）。在有噪声的情况下，该RIC上限沿着连同对最小信号输入幅度的要求是已知的，以保证精确的支撑识别。在本文中，我们表明，在噪声的存在下，一个放松的RIC上限δK+1 <;（<$（4K + 1）- 1/2K）连同放松的最小信号输入幅度的要求，足以实现完美的支持识别使用OMP。在无噪声的情况下，我们的结果表明，这样一个放松的RIC上界可以保证在K次迭代中准确地恢复支持度：这缩小了迄今为止最好的已知界δK+1 <;（1/K（+ 1））和最终性能保证δK+1 =（1/（K））之间的差距。我们的方法依赖于一个新建立的近正交条件，其特征在于通过压缩时两个正交稀疏向量之间的可实现的角度，因此，更好地利用压缩空间的几何知识。所提出的近正交性条件也可以被利用来导出用于其他两个压缩感知问题中的信号重构的较少限制的充分条件，即，压缩域干扰消除和通过子空间追踪算法的支持识别。
A sufficient condition reported very recently for perfect recovery of a K-sparse vector via orthogonal matching pursuit (OMP) in K iterations (when there is no noise) is that the restricted isometry constant (RIC) of the sensing matrix satisfies δK+1 <; (1/√(K) + 1). In the noisy case, this RIC upper bound along with a requirement on the minimal signal entry magnitude is known to guarantee exact support identification. In this paper, we show that, in the presence of noise, a relaxed RIC upper bound δK+1 <; (√(4K + 1) - 1/2K) together with a relaxed requirement on the minimal signal entry magnitude suffices to achieve perfect support identification using OMP. In the noiseless case, our result asserts that such a relaxed RIC upper bound can ensure exact support recovery in K iterations: this narrows the gap between the so far best known bound δK+1 <; (1/√(K( + 1)) and the ultimate performance guarantee δK+1 = (1/(K)). Our approach relies on a newly established near orthogonality condition, characterized via the achievable angles between two orthogonal sparse vectors upon compression, and, thus, better exploits the knowledge about the geometry of the compressed space. The proposed near orthogonality condition can be also exploited to derive less restricted sufficient conditions for signal reconstruction in two other compressive sensing problems, namely, compressive domain interference cancellation and support identification via the subspace pursuit algorithm.