@article{6662,
abstract = {In phase retrieval, we want to recover an unknown signal π₯ββπ from n quadratic measurements of the form π¦π=|β¨ππ,π₯β©|2+π€π, where ππββπ are known sensing vectors and π€π is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator π₯^(π¦) that is positively correlated with the signal π₯? We consider the case of Gaussian vectors πππ. We prove thatβin the high-dimensional limitβa sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For πβ€πβπ(π), no estimator can do significantly better than random and achieve a strictly positive correlation. For πβ₯π+π(π), a simple spectral estimator achieves a positive correlation. Surprisingly, numerical simulations with the same spectral estimator demonstrate promising performance with realistic sensing matrices. Spectral methods are used to initialize non-convex optimization algorithms in phase retrieval, and our approach can boost the performance in this setting as well. Our impossibility result is based on classical information-theoretic arguments. The spectral algorithm computes the leading eigenvector of a weighted empirical covariance matrix. We obtain a sharp characterization of the spectral properties of this random matrix using tools from free probability and generalizing a recent result by Lu and Li. Both the upper bound and lower bound generalize beyond phase retrieval to measurements π¦π produced according to a generalized linear model. As a by-product of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.},
author = {Mondelli, Marco and Montanari, Andrea},
issn = {1615-3383},
journal = {Foundations of Computational Mathematics},
number = {3},
pages = {703--773},
publisher = {Springer},
title = {{Fundamental limits of weak recovery with applications to phase retrieval}},
doi = {10.1007/s10208-018-9395-y},
volume = {19},
year = {2019},
}