Papers › Sparse Phase Retrieval via Truncated Amplitude Flow
Sparse Phase Retrieval via Truncated Amplitude Flow
Gang Wang, Liang Zhang, Georgios B. Giannakis, Mehmet Akcakaya, Jie Chen
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This paper develops a novel algorithm, termed \emph{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 \emph{NP-hard} in general and emerges in many science and engineering applications. Upon formulating sparse PR as an amplitude-based 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))x₂, SPARTA recovers the signal exactly (up to a global unimodular constant) from about k²logn random Gaussian measurements with high probability. Furthermore, SPARTA incurs computational complexity on the order of k²nlogn 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.
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