Scalable Solvers 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 \boldsymbol {x} from m “phaseless” quadratic equations of the form \psi _i=|\langle \boldsymbol a_i,\boldsymbol x\rangle | . 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 are useful when n is large. When \lbrace \boldsymbol a_i\rbrace i=1^m are independent Gaussian, STAF provably recovers exactly any \boldsymbol x\in \mathbbR^n 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 \lbrace \boldsymbol {a}_i\rbrace vectors demonstrate that STAF empirically reconstructs any \boldsymbol x\in \mathbbR^n 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. [ABSTRACT FROM PUBLISHER]