QUASI-LINEAR COMPRESSED SENSING
QUASI-LINEAR COMPRESSED SENSING
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DOI：
10.1137/130929928
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发表时间：
2014-01-01
影响因子：
1.6
通讯作者：
Sigl, Juliane
中科院分区：
文献类型：
--
作者：
Ehler, Martin;Fornasier, Massimo;Sigl, Juliane
受现实生活中的重要应用，特别是稀疏相位检索和稀疏脉动频率检测在asteroseismology的启发，我们研究了压缩感知的一般框架，其中的测量是准线性的。我们制定自然的推广著名的限制等距属性（RIP）对非线性测量，这使我们能够证明稀疏信号的唯一可识别性以及恢复算法的收敛性，以有效地计算它们。我们发现，对于某些随机准线性测量，包括经典RIP矩阵的Lipschitz扰动和随机投影的相位恢复，建议的限制等距属性具有很高的概率。我们分析了广义正交最小二乘（OLS）的假设下，要恢复的信号条目的幅度衰减很快。贪婪是好的，因为我们表明，这种算法有效地执行相位恢复和asteroseismology。对于信号上的衰减假设不一定成立的情况下，我们提出了两种替代算法，这是众所周知的迭代硬阈值和软阈值的自然概括。虽然这些算法很少成功的上述应用程序，我们表现出强有力的恢复保证准线性测量，这是RIP矩阵的Lipschitz扰动。
Inspired by significant real-life applications, particularly sparse phase retrieval and sparse pulsation frequency detection in asteroseismology, we investigate a general framework for compressed sensing, where the measurements are quasi-linear. We formulate natural generalizations of the well-known restricted isometry property (RIP) toward nonlinear measurements, which allow us to prove both unique identifiability of sparse signals as well as the convergence of recovery algorithms to compute them efficiently. We show that for certain randomized quasi-linear measurements, including Lipschitz perturbations of classical RIP matrices and phase retrieval from random projections, the proposed restricted isometry properties hold with high probability. We analyze a generalized orthogonal least squares (OLS) under the assumption that magnitudes of signal entries to be recovered decay quickly. Greed is good again, as we show that this algorithm performs efficiently in phase retrieval and asteroseismology. For situations where the decay assumption on the signal does not necessarily hold, we propose two alternative algorithms, which are natural generalizations of the well-known iterative hard- and soft-thresholding. While these algorithms are rarely successful for the mentioned applications, we show their strong recovery guarantees for quasi-linear measurements which are Lipschitz perturbations of RIP matrices.