Hadamard Wirtinger Flow for Sparse Phase Retrieval

Phase retrieval, the problem of reconstructing a signal from the (squared) magnitude of its Fourier (or any linear) transform, arises in many fields of science and engineering. Such a task is naturally involved in applications such as crystallography (Millane, 1990) and diffraction imaging (Bunk et al., 2007), where optical sensors are able to measure the intensity, but not the phase of a light wave. Due to the loss of phase information, the one-dimensional Fourier phase retrieval problem is ill-posed in general. Common approaches to overcome this ill-posedness include using prior information such as non-negativity, sparsity and the signal's magnitude (Fienup, 1982; Jaganathan et al., 2016), or introducing redundancy into the measurements by oversampling random Gaussian measurements or coded diffraction patterns (Candès et al., 2015; Chen and Candès, 2015). In many applications, the underlying signal is naturally sparse (Jaganathan et al., 2016). A wide range of algorithms has been devised for phase retrieval with a sparse signal, including alternating minimization (SparseAltMinPhase) (Netrapalli et al., 2015), non-convex optimization based approaches such as thresholded Wirtinger flow (TWF) (Cai et al., 2016), sparse truncated amplitude flow (SPARTA) (Wang et al., 2018), compressive reweighted amplitude flow (CRAF) (Zhang et al., 2018) and sparse Wirtinger flow (SWF) (Yuan et al., 2019), and convex relaxation methods such as compressive phase retrieval via lifting (CPRL) (Ohlsson et al., 2012) and SparsePhaseMax (Hand and Voroninski, 2016). Other approaches to sparse phase retrieval include the greedy algorithm GESPAR (Schechtman et al., 2014), an algorithm based on generalized