## Abstract

A novel approach termed stochastic truncated amplitude flow (STAF) is developed to reconstruct an unknown n-dimensional real-/complex-valued signal x fromm "phaseless" quadratic equations of the form Ψ_{i} = |ai, x|. 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 {a_{i}}^{m} i=1 are independent Gaussian, STAF provably recovers exactly any x ϵ R^{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 {ai} vectors demonstrate that STAF empirically reconstructs any x ϵ R^{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.

Original language | English (US) |
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Article number | 7815432 |

Pages (from-to) | 1961-1974 |

Number of pages | 14 |

Journal | IEEE Transactions on Signal Processing |

Volume | 65 |

Issue number | 8 |

DOIs | |

State | Published - Apr 15 2017 |

### Bibliographical note

Funding Information:This work was supported in part by NSF under Grant 1500713 and Grant 1514056.

Publisher Copyright:

© 2017 IEEE.

## Keywords

- Kaczmarz algorithm
- Nonconvex optimization
- phase retrieval
- variance reduction