SPARTA: Sparse phase retrieval via Truncated Amplitude flow

A linear-time algorithm termed SPARse Truncated Amplitude flow (SPARTA) is developed for the phase retrieval (PR) of sparse signals. Upon formulating the sparse PR as a non-convex empirical loss minimization task, SPARTA emerges as an iterative solver consisting of two components: s1) a sparse orthogonality-promoting initialization leveraging support recovery and principal component analysis; and, s2) a series of refinements by hard thresholding based truncated gradient iterations. SPARTA is simple, scalable, and fast. It recovers any k-sparse n-dimensional signal (k ≪ n) of large enough minimum (in modulus) nonzero entries from about k² log n measurements with high probability; this is achieved at computational complexity of order k²n log n, improving upon the state-of-the-art by at least a factor of k. SPARTA is robust against bounded additive noise. Simulated tests corroborate the merits of SPARTA relative to existing alternatives.