Quasi-linear Compressed Sensing

Abstract

Abstract. Inspired by significant real-life applications, in particular, 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) towards 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 fast. 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.Key words. compressed sensing, restricted isometry property, greedy algorithm, quasi-linear, iterative thresholding AMS subject classifications. 94A20, 47J25, 15B521. Introduction. Compressed sensing addresses the problem of recovering nearly-sparse signals from vastly incomplete measurements [11,12,14,15,21]. By using the prior assumptions on the signal, the number of measurements can be well below the Shannon sampling rate and effective reconstruction algorithms are available. The standard compressed sensing approach deals with linear measurements. The success of signal recovery algorithms often relies on the so-called Restricted Isometry Property (RIP) [12,15,27,35,38,39], which is a near-identity spectral property of small submatrices of the measurement Gramian. The RIP condition is satisfied with high probability and nearly optimal number of measurements for a large class of random measurements [3,4,14,35,38], which explains the popularity of all sorts of random sensing approaches. The most effective recovery algorithms are based either on a greedy approach or on variational models, such as 1 -norm minimization, leading to suitable iterative thresholded gradient descent methods. In the literature of mathematical signal processing, greedy algorithms for sparse recovery originate from the so-called Matching Pursuit [33], although several predecessors were well-known in other communities. Among astronomers and asteroseismologists, for instance, Orthogonal Least Squares (OLS) [31] was already in use in the '60s for the detection of significant frequencies of star light-spectra (the so-called prewhitening) [...

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