### Abstract

This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form y_{i}=|{a_{i}, x}|^{2}, where a-_{i}'s are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adapts the amplitude-based empirical loss function and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and a_{i}'s are real valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.

Original language | English (US) |
---|---|

Article number | 8049465 |

Pages (from-to) | 773-794 |

Number of pages | 22 |

Journal | IEEE Transactions on Information Theory |

Volume | 64 |

Issue number | 2 |

DOIs | |

State | Published - Feb 2018 |

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### Keywords

- Amplitude-based cost function
- Linear convergence to global minimum
- Nonconvex optimization
- Orthogonality-promoting initialization
- Phase retrieval
- Truncated gradient

### Cite this

*IEEE Transactions on Information Theory*,

*64*(2), 773-794. [8049465]. https://doi.org/10.1109/TIT.2017.2756858

**Solving systems of random quadratic equations via truncated amplitude flow.** / Wang, Gang; Giannakis, Georgios B.; Eldar, Yonina C.

Research output: Contribution to journal › Article

*IEEE Transactions on Information Theory*, vol. 64, no. 2, 8049465, pp. 773-794. https://doi.org/10.1109/TIT.2017.2756858

}

TY - JOUR

T1 - Solving systems of random quadratic equations via truncated amplitude flow

AU - Wang, Gang

AU - Giannakis, Georgios B.

AU - Eldar, Yonina C.

PY - 2018/2

Y1 - 2018/2

N2 - This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi=|{ai, x}|2, where a-i's are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adapts the amplitude-based empirical loss function and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai's are real valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.

AB - This paper presents a new algorithm, termed truncated amplitude flow (TAF), to recover an unknown vector x from a system of quadratic equations of the form yi=|{ai, x}|2, where a-i's are given random measurement vectors. This problem is known to be NP-hard in general. We prove that as soon as the number of equations is on the order of the number of unknowns, TAF recovers the solution exactly (up to a global unimodular constant) with high probability and complexity growing linearly with both the number of unknowns and the number of equations. Our TAF approach adapts the amplitude-based empirical loss function and proceeds in two stages. In the first stage, we introduce an orthogonality-promoting initialization that can be obtained with a few power iterations. Stage two refines the initial estimate by successive updates of scalable truncated generalized gradient iterations, which are able to handle the rather challenging nonconvex and nonsmooth amplitude-based objective function. In particular, when vectors x and ai's are real valued, our gradient truncation rule provably eliminates erroneously estimated signs with high probability to markedly improve upon its untruncated version. Numerical tests using synthetic data and real images demonstrate that our initialization returns more accurate and robust estimates relative to spectral initializations. Furthermore, even under the same initialization, the proposed amplitude-based refinement outperforms existing Wirtinger flow variants, corroborating the superior performance of TAF over state-of-the-art algorithms.

KW - Amplitude-based cost function

KW - Linear convergence to global minimum

KW - Nonconvex optimization

KW - Orthogonality-promoting initialization

KW - Phase retrieval

KW - Truncated gradient

UR - http://www.scopus.com/inward/record.url?scp=85030784455&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85030784455&partnerID=8YFLogxK

U2 - 10.1109/TIT.2017.2756858

DO - 10.1109/TIT.2017.2756858

M3 - Article

AN - SCOPUS:85030784455

VL - 64

SP - 773

EP - 794

JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

SN - 0018-9448

IS - 2

M1 - 8049465

ER -