Universality of Linearized Message Passing for Phase Retrieval With Structured Sensing Matrices

In the phase retrieval problem one seeks to recover an unknown <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> dimensional signal vector <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></in...

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Veröffentlicht in:IEEE transactions on information theory Jg. 68; H. 11; S. 7545 - 7574
Hauptverfasser: Dudeja, Rishabh, Bakhshizadeh, Milad
Format: Journal Article
Sprache:Englisch
Veröffentlicht: New York IEEE 01.11.2022
The Institute of Electrical and Electronics Engineers, Inc. (IEEE)
Schlagworte:
Algorithms
Approximation algorithms
Columns (structural)
Discrete Fourier transforms
Evolution
Extraterrestrial measurements
Fourier matrix
Hadamard-Walsh matrix
Heuristic algorithms
Linearization
Message passing
message passing algorithms
Phase measurement
Phase retrieval
Sampling
Sensors
universality
ISSN:0018-9448, 1557-9654
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Zusammenfassung:In the phase retrieval problem one seeks to recover an unknown <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> dimensional signal vector <inline-formula> <tex-math notation="LaTeX">\mathbf {x} </tex-math></inline-formula> from <inline-formula> <tex-math notation="LaTeX">m </tex-math></inline-formula> measurements of the form <inline-formula> <tex-math notation="LaTeX">y_{i} = |(\mathbf {A} \mathbf {x})_{i}| </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">\mathbf {A} </tex-math></inline-formula> denotes the sensing matrix. Many algorithms for this problem are based on approximate message passing. For these algorithms, it is known that if the sensing matrix <inline-formula> <tex-math notation="LaTeX">\mathbf {A} </tex-math></inline-formula> is generated by sub-sampling <inline-formula> <tex-math notation="LaTeX">n </tex-math></inline-formula> columns of a uniformly random (i.e., Haar distributed) orthogonal matrix, in the high dimensional asymptotic regime (<inline-formula> <tex-math notation="LaTeX">m,n \rightarrow \infty, n/m \rightarrow \kappa </tex-math></inline-formula>), the dynamics of the algorithm are given by a deterministic recursion known as the state evolution. For a special class of linearized message-passing algorithms, we show that the state evolution is universal: it continues to hold even when <inline-formula> <tex-math notation="LaTeX">\mathbf {A} </tex-math></inline-formula> is generated by randomly sub-sampling columns of the Hadamard-Walsh matrix, if the signal is drawn from a Gaussian prior.
Bibliographie:ObjectType-Article-1
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ISSN:0018-9448
1557-9654
DOI:10.1109/TIT.2022.3182018