Sparse Phase Retrieval via Truncated Amplitude Flow

Abstract

This paper develops a novel algorithm, termed SPARse Truncated Amplitude flow (SPARTA), to reconstruct a sparse signal from a small number of magnitude-only measurements. It deals with what is also known as sparse phase retrieval (PR), which is NP-hard in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitudebased nonconvex optimization task, SPARTA works iteratively in two stages: In stage one, the support of the underlying sparse signal is recovered using an analytically well-justified rule, and subsequently a sparse orthogonality-promoting initialization is obtained via power iterations restricted on the support; and, in the second stage, the initialization is successively refined by means of hard thresholding based gradient-type iterations. SPARTA is a simple yet effective, scalable, and fast sparse PR solver. On the theoretical side, for any n-dimensional k-sparse (k n) signal x with minimum (in modulus) nonzero entries on the order of (1/ √ k) x 2, SPARTA recovers the signal exactly (up to a global unimodular constant) from about k 2 log n random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of k 2 n log n with total runtime proportional to the time required to read the data, which improves upon the state-of-the-art by at least a factor of k. Finally, SPARTA is robust against additive noise of bounded support. Extensive numerical tests corroborate markedly improved recovery performance and speedups of SPARTA relative to existing alternatives. and ptychography, astronomy, optics, as well as array and coherent diffraction imaging. In these settings, optical sensors and detectors such as charge-coupled device cameras, photosensitive films, and human eyes record only the intensity (squared magnitude) of a light wave, but not the phase. In particular, solution to PR has led to significant accomplishments, including the discovery in 1953 of DNA double helical structure from diffraction patterns, and the characterization of aberrations in the Hubble Space Telescope from measured point spread functions [1]. Due to the absence of Fourier phase information, the one-dimensional (1D) Fourier PR problem is generally ill-posed. It can be shown that there are in fact exponentially many non-equivalent solutions beyond trivial ambiguities in the 1D PR case [2]. A common approach to overcome this ill-posedness is exploiting additional information on the unknown signal such as non-negativity, sparsity, or bounded magnitude [3,4,5]. Other viable solutions consist of introducing redundancy into the measurement transforming system to obtain over-sampled and short-time Fourier transform (STFT) measurements [6], random Gaussian measurements [7,8,9], and coded diffraction patterns using structured illumination and random masks [10,11,7], just to name a few; see [10] for contemporary reviews on the theory and practice of PR.Past PR approaches can be mainly categorized as convex and nonconvex ones. A popular class of ...

Provable Low Rank Phase Retrieval

“…The second inequality used (16), while the third used incoherence of b k 's. The last inequality used incoherence of b k 's and of p k 's (proved above) and X * F ≥ √ rσ * min .…”

“…In fact it can be understood as phaseless compressive sensing. Provably correct sparse PR approaches include convex relaxation approaches such as 1 -PhaseLift [14]; older combinatorial methods [48]; and a series of fast iterative approaches: (i) AltMinSparse [6], (ii) Sparse Truncated Amplitude Flow (SPARTA) [16], (iii) Thresholded WF [17] and CoPRAM [18]. The first two fast nonconvex approaches -AltMinSparse and SPARTA -needed to assume a lower bound on the minimum nonzero entry of x.…”

“…where we used BB = I r . Now we use the above and (16) to show that it also concentrates around zero. Let X ik = a ik W b k and Y ik = a ik U * p k .…”

Alternating Phase Projected Gradient Descent with Generative Priors for Solving Compressive Phase Retrieval

“…Some of the convex approaches for sparse phase retrieval include [20,21,22,23]. Similarly, nonconvex approaches for sparse phase retrieval includes [9,19,14].…”

Randomized Block Frank–Wolfe for Convergent Large-Scale Learning

“…Typically, solving a constrained linear optimization is considerably easier than finding the aforementioned projections per iteration. The resulting savings benefit diverse large-scale learning tasks, including matrix completion [6], multi-class classification [7], image reconstruction [7], structural support vector machines (SVMs) [8], particle filtering [9], sparse phase retrieval [10], [11], and scheduling electric vehicle (EV) charging [12].…”