Fundamental Limits of Weak Recovery with Applications to Phase Retrieval
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In phase retrieval we want to recover an unknown signal x∈ C^d from n quadratic measurements of the form y_i = |〈 a_i, x〉|^2+w_i where a_i∈ C^d are known sensing vectors and w_i is measurement noise. We ask the following weak recovery question: what is the minimum number of measurements n needed to produce an estimator x̂( y) that is positively correlated with the signal x? We consider the case of Gaussian vectors a_i. We prove that - in the high-dimensional limit - a sharp phase transition takes place, and we locate the threshold in the regime of vanishingly small noise. For n< d-o(d) no estimator can do significantly better than random and achieve a strictly positive correlation. For n> d+o(d) 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-theory 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 and lower bound generalize beyond phase retrieval to measurements y_i produced according to a generalized linear model. As a byproduct of our analysis, we compare the threshold of the proposed spectral method with that of a message passing algorithm.
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