7 research outputs found
A Deterministic Theory for Exact Non-Convex Phase Retrieval
In this paper, we analyze the non-convex framework of Wirtinger Flow (WF) for
phase retrieval and identify a novel sufficient condition for universal exact
recovery through the lens of low rank matrix recovery theory. Via a perspective
in the lifted domain, we show that the convergence of the WF iterates to a true
solution is attained geometrically under a single condition on the lifted
forward model. As a result, a deterministic relationship between the accuracy
of spectral initialization and the validity of {the regularity condition} is
derived. In particular, we determine that a certain concentration property on
the spectral matrix must hold uniformly with a sufficiently tight constant.
This culminates into a sufficient condition that is equivalent to a restricted
isometry-type property over rank-1, positive semi-definite matrices, and
amounts to a less stringent requirement on the lifted forward model than those
of prominent low-rank-matrix-recovery methods in the literature. We
characterize the performance limits of our framework in terms of the tightness
of the concentration property via novel bounds on the convergence rate and on
the signal-to-noise ratio such that the theoretical guarantees are valid using
the spectral initialization at the proper sample complexity.Comment: In Revision for IEEE Transactions on Signal Processin
Phase retrieval with random Gaussian sensing vectors by alternating projections
We consider a phase retrieval problem, where we want to reconstruct a
-dimensional vector from its phaseless scalar products with sensing
vectors. We assume the sensing vectors to be independently sampled from complex
normal distributions. We propose to solve this problem with the classical
non-convex method of alternating projections. We show that, when for
large enough, alternating projections succeed with high probability,
provided that they are carefully initialized. We also show that there is a
regime in which the stagnation points of the alternating projections method
disappear, and the initialization procedure becomes useless. However, in this
regime, has to be of the order of . Finally, we conjecture from our
numerical experiments that, in the regime , there are stagnation
points, but the size of their attraction basin is small if is large
enough, so alternating projections can succeed with probability close to
even with no special initialization