Fundamental Limits of Weak Recovery with Applications to Phase Retrieval

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...

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Veröffentlicht in:	Foundations of computational mathematics Jg. 19; H. 3; S. 703 - 773
Hauptverfasser:	Mondelli, Marco, Montanari, Andrea
Format:	Journal Article
Sprache:	Englisch
Veröffentlicht:	New York Springer US 01.06.2019 Springer Nature B.V
Schlagworte:	Algorithms Applications of Mathematics Computational geometry Computer Science Computer simulation Convexity Correlation Covariance matrix Economics Eigenvectors Empirical analysis Generalized linear models Information theory Linear and Multilinear Algebras Lower bounds Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Message passing Noise measurement Numerical Analysis Optimization Phase retrieval Phase transitions Recovery Spectra Spectral methods Statistical models Upper bounds Spectral initialization Second-moment method 91E40 68T05 Phase retrieval 68Q32 Mutual information Phase transition
ISSN:	1615-3375, 1615-3383
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Abstract	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 x ^ ( 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-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.
AbstractList	In phase retrieval, we want to recover an unknown signal x∈Cd from n quadratic measurements of the form yi=\|⟨ai,x⟩\|2+wi, where ai∈Cd are known sensing vectors and wi is measurement noise. We ask the following weak recovery question: What is the minimum number of measurements n needed to produce an estimator x^(y) that is positively correlated with the signal x? We consider the case of Gaussian vectors ai. 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-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 yi 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. 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 x ^ ( 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-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.
Author	Mondelli, Marco Montanari, Andrea
Author_xml	– sequence: 1 givenname: Marco surname: Mondelli fullname: Mondelli, Marco email: mondelli@stanford.edu organization: Department of Electrical Engineering, Stanford University – sequence: 2 givenname: Andrea surname: Montanari fullname: Montanari, Andrea organization: Department of Electrical Engineering, Stanford University, Department of Statistics, Stanford University
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Snippet	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... In phase retrieval, we want to recover an unknown signal x∈Cd from n quadratic measurements of the form yi=\|⟨ai,x⟩\|2+wi, where ai∈Cd are known sensing vectors...
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SubjectTerms	Algorithms Applications of Mathematics Computational geometry Computer Science Computer simulation Convexity Correlation Covariance matrix Economics Eigenvectors Empirical analysis Generalized linear models Information theory Linear and Multilinear Algebras Lower bounds Math Applications in Computer Science Mathematics Mathematics and Statistics Matrix Theory Message passing Noise measurement Numerical Analysis Optimization Phase retrieval Phase transitions Recovery Spectra Spectral methods Statistical models Upper bounds
Title	Fundamental Limits of Weak Recovery with Applications to Phase Retrieval
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