Solving Large-scale Systems of Random Quadratic Equations via Stochastic Truncated Amplitude Flow

A novel approach termed stochastic truncated amplitude flow (STAF) is developed to reconstruct an unknown n-dimensional real-/complex-valued signal bmx from m `phaseless' quadratic equations of the form psii=|langlebmai,bmxrangle|. This problem, also known as phase retrieval from magnitude-only information, is NP-hard in general. Adopting an amplitude-based nonconvex formulation, STAF leads to an iterative solver comprising two stages: s1) Orthogonality-promoting initialization through a stochastic variance reduced gradient algorithm; and, s2) A series of iterative refinements of the initialization using stochastic truncated gradient iterations. Both stages involve a single equation per iteration, thus rendering STAF a simple, scalable, and fast approach amenable to large-scale implementations that is useful when n is large. When bmaii=1m are independent Gaussian, STAF provably recovers exactly any bmx∈ℝn exponentially fast based on order of n quadratic equations. STAF is also robust in the presence of additive noise of bounded support. Simulated tests involving real Gaussian bmai vectors demonstrate that STAF empirically reconstructs any bmx∈ℝn exactly from about 2.3n magnitude-only measurements, outperforming state-of-the-art approaches and narrowing the gap from the information-theoretic number of equations m=2n-1. Extensive experiments using synthetic data and real images corroborate markedly improved performance of STAF over existing alternatives.