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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Bibliographic Details
Published in:Foundations of computational mathematics Vol. 19; no. 3; pp. 703 - 773
Main Authors: Mondelli, Marco, Montanari, Andrea
Format: Journal Article
Language:English
Published: New York Springer US 01.06.2019
Springer Nature B.V
Subjects:
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
Online Access:Get full text
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Summary: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.
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ISSN:1615-3375
1615-3383
DOI:10.1007/s10208-018-9395-y