Quasi-linear Compressed Sensing

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 situ...

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