Scalable Incremental Nonconvex Optimization Approach for Phase Retrieval
from Minimal Measurements
release_bprsgq2mhzdcnmroab6ksrqcte
by
Ji Li, Jian-Feng Cai, Hongkai Zhao
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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