Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval from Minimal Measurements release_bprsgq2mhzdcnmroab6ksrqcte

by Ji Li, Jian-Feng Cai, Hongkai Zhao

Released as a article .

2018  

Abstract

We aim to find a solution x∈R^n/C^n to a system of quadratic equations of the form b_i=〈a_i,x〉^2, i=1,2,...,m, e.g., the well-known phase retrieval problem, which is generally NP-hard. It has been proved that the number m = 2n-1 of generic random measurement vectors a_i∈R^n is sufficient and necessary for uniquely determining the n-length real vector x up to a global sign. The uniqueness theory, however, does not provide a construction or characterization of this unique solution. As opposed to the recent nonconvex state-of-the-art solvers, we revert to the convex relaxation semidefinite programming (SDP) approach and propose to indirectly minimize the convex objective by successive and incremental nonconvex optimization, termed as IncrePR, to overcome the excessive computation cost of typical SDP solvers. IncrePR avoids sensitive dependence of initialization of nonconvex approaches and achieves global convergence, which makes it also promising for more general models and measurements. For real Gaussian model, IncrePR achieves perfect recovery from m=2n-1 noiseless measurement and the recovery is stable from noisy measurement. When applying IncrePR for structured (non-Gaussian) measurements, such as transmission matrix and oversampling Fourier measurement, it can also locate a reconstruction close to true reconstruction with few measurements. Extensive numerical tests show that IncrePR outperforms other state-of-the-art methods in the sharpest phase transition of perfect recovery for Gaussian model and the best reconstruction quality for other non-Gaussian models, in particular Fourier phase retrieval.
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Date   2018-07-19
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arXiv  1807.05499v2
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