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• W2964130120 abstract "In phase retrieval, we want to recover an unknown signal $${{varvec{x}}}in {{mathbb {C}}}^d$$ from n quadratic measurements of the form $$y_i = |langle {{varvec{a}}}_i,{{varvec{x}}}rangle |^2+w_i$$ , where $${{varvec{a}}}_iin {{mathbb {C}}}^d$$ 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 $${hat{{{varvec{x}}}}}({{varvec{y}}})$$ that is positively correlated with the signal $${{varvec{x}}}$$ ? We consider the case of Gaussian vectors $${{varvec{a}}}_i$$ . 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 $$nle d-o(d)$$ , no estimator can do significantly better than random and achieve a strictly positive correlation. For $$nge d+o(d)$$ , 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-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 $$y_i$$ 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." @default.
• W2964130120 created "2019-07-30" @default.
• W2964130120 creator A5011999109 @default.
• W2964130120 creator A5061114394 @default.
• W2964130120 date "2018-09-04" @default.
• W2964130120 modified "2023-09-25" @default.
• W2964130120 title "Fundamental Limits of Weak Recovery with Applications to Phase Retrieval" @default.
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• W2964130120 doi "https://doi.org/10.1007/s10208-018-9395-y" @default.
• W2964130120 hasPublicationYear "2018" @default.
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